// Numbas version: finer_feedback_settings
{"name": "MPR Practice Questions", "metadata": {"description": "<p>A set of practice questions for primary teaching students studying Mathematical Patterns and Relationships</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": true, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Place Value", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Face, Place and Actual Value (integers)", "Face, Place and Actual Value (decimals)"], "variable_overrides": [[], []], "questions": [{"name": "Merryn's copy of Face, Place and Actual Value (base 10 integers)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Useful for a review of the base 10 number system before introducing different bases and also just ensuring students understand how the base 10 system works.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>In our usual number <g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\">system</g> we only have ten digits (0 to 9). We call this the base 10 system. We can use&nbsp;these ten digits together to represent numbers larger than 9. When we&nbsp;do this the place we put the numbers is important, for example, $321$ uses the same digits as $123$ but it is clearly not the same number. This question&nbsp;tests your understanding of the face, place and actual values in the base 10 number system.</p>", "advice": "<p>a) The&nbsp;<strong>face value</strong> of a digit is simply the digit itself. It is all about what the digit looks like.</p>\n<p>For example, the face value of $2$ in the number $1234$ is just $2$.</p>\n<p>b) The&nbsp;<strong>place value</strong> of a digit is the 'value' of the 'column' it is written in. It is all about where the digit is. Recall we have a thousands column, a hundreds column, a tens column and a units/ones column. The place value for something in the thousands column is $1000$, the place value for something in the hundreds column is $100$, the place value for something in the tens column is $10$ and the place value for something in the units/ones column is $1$.</p>\n<p>For example, the place value of the $2$ in the number $1234$ is $100$.</p>\n<p>c) The&nbsp;<strong>actual value</strong> of a digit is the face value times the place value. It is all about what that digit actually adds/brings to the number, what it is actually worth or what it actually represents.</p>\n<p>For example, the actual value of the digit $2$ in the number $1234$ is $200$ (since the $2$ in $1234$ actually stands for $200$).</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"int": {"name": "int", "group": "Ungrouped variables", "definition": "latex(join(listOfDigits,''))", "description": "", "templateType": "anything", "can_override": false}, "thousandsA": {"name": "thousandsA", "group": "Ungrouped variables", "definition": "thousandsF*b^3", "description": "", "templateType": "anything", "can_override": false}, "thousandsF": {"name": "thousandsF", "group": "try this", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "valueinbase10": {"name": "valueinbase10", "group": "Ungrouped variables", "definition": "thousandsA+hundredsA+tensA+units", "description": "", "templateType": "anything", "can_override": false}, "tensA": {"name": "tensA", "group": "Ungrouped variables", "definition": "tensF*b", "description": "", "templateType": "anything", "can_override": false}, "otherDigits": {"name": "otherDigits", "group": "try this", "definition": "shuffle(0..9 except thousandsF)[0..3]", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "10", "description": "", "templateType": "anything", "can_override": false}, "hundredsF": {"name": "hundredsF", "group": "Ungrouped variables", "definition": "listofDigits[1]", "description": "", "templateType": "anything", "can_override": false}, "tensF": {"name": "tensF", "group": "Ungrouped variables", "definition": "listOfDigits[2]", "description": "", "templateType": "anything", "can_override": false}, "hundredsA": {"name": "hundredsA", "group": "Ungrouped variables", "definition": "hundredsF*b^2", "description": "", "templateType": "anything", "can_override": false}, "listOfDigits": {"name": "listOfDigits", "group": "try this", "definition": "[thousandsF]+otherDigits", "description": "<p>listofDigits</p>", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "listOfDigits[3]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["int", "thousandsA", "hundredsF", "hundredsA", "tensF", "tensA", "units", "b", "valueinbase10"], "variable_groups": [{"name": "try this", "variables": ["thousandsF", "otherDigits", "listOfDigits"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>face value</strong> of the following digits from the base $\\var{b}$ number $\\var{int}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{thousandsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredsF}$</td>\n<td style=\"text-align: center;\">$\\var{tensF}$</td>\n<td style=\"text-align: center;\">$\\var{units}$</td>\n</tr>\n<tr>\n<td>Face Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "thousandsF", "maxValue": "thousandsF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "hundredsF", "maxValue": "hundredsF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tensF", "maxValue": "tensF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "units", "maxValue": "units", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>place value</strong> of the following digits from the base $\\var{b}$ number $\\var{int}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{thousandsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredsF}$</td>\n<td style=\"text-align: center;\">$\\var{tensF}$</td>\n<td style=\"text-align: center;\">$\\var{units}$</td>\n</tr>\n<tr>\n<td>Place Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b^3", "maxValue": "b^3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b^2", "maxValue": "b^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b", "maxValue": "b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>actual value</strong> of the following digits from the base $\\var{b}$ number $\\var{int}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{thousandsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredsF}$</td>\n<td style=\"text-align: center;\">$\\var{tensF}$</td>\n<td style=\"text-align: center;\">$\\var{units}$</td>\n</tr>\n<tr>\n<td>Actual Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "thousandsA", "maxValue": "thousandsA", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "hundredsA", "maxValue": "hundredsA", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tensA", "maxValue": "tensA", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "units", "maxValue": "units", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Face, Place and Actual Value (decimals)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Useful for a review of the base 10 number system before introducing different bases and also just ensuring students understand how the base 10 system works.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>In our usual number <g class=\"gr_ gr_11 gr-alert gr_gramm gr_inline_cards gr_run_anim Punctuation only-ins replaceWithoutSep\" id=\"11\" data-gr-id=\"11\">system</g> we only have ten digits (0 to 9). We call this the base 10 system. We can use&nbsp;these ten digits together to represent numbers larger than 9. When we&nbsp;do this the place we put the numbers is important, for example, $321$ uses the same digits as $123$ but it is clearly not the same number. This question&nbsp;tests your understanding of the face, place and actual values in the decimal number system.</p>", "advice": "<p></p>\n<p></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"thousandthsF": {"name": "thousandthsF", "group": "try this", "definition": "random(1..9 except [tenthsF,hundredthsF])", "description": "", "templateType": "anything", "can_override": false}, "tenthsF": {"name": "tenthsF", "group": "try this", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "try this", "definition": "10", "description": "", "templateType": "anything", "can_override": false}, "tenthsA": {"name": "tenthsA", "group": "try this", "definition": "tenthsF/b", "description": "", "templateType": "anything", "can_override": false}, "hundredthsA": {"name": "hundredthsA", "group": "try this", "definition": "hundredthsF/b^2", "description": "", "templateType": "anything", "can_override": false}, "dec": {"name": "dec", "group": "try this", "definition": "tenthsA+hundredthsA+thousandthsA", "description": "", "templateType": "anything", "can_override": false}, "thousandthsA": {"name": "thousandthsA", "group": "try this", "definition": "thousandthsF/b^3", "description": "", "templateType": "anything", "can_override": false}, "hundredthsF": {"name": "hundredthsF", "group": "try this", "definition": "random(1..9 except tenthsF)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "try this", "variables": ["tenthsF", "hundredthsF", "thousandthsF", "tenthsA", "hundredthsA", "thousandthsA", "dec", "b"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>face value</strong> of the following digits from the&nbsp;decimal&nbsp;$\\var{dec}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{tenthsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredthsF}$</td>\n<td style=\"text-align: center;\">$\\var{thousandthsF}$</td>\n</tr>\n<tr>\n<td>Face Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "stepsPenalty": "3", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The&nbsp;<strong>face value</strong> of a digit is simply the digit itself. It is all about what the digit looks like.</p>\n<p>For example, the face value of $3$ in the number $1.234$ is just $3$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tenthsF", "maxValue": "tenthsF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "hundredthsF", "maxValue": "hundredthsF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "thousandthsF", "maxValue": "thousandthsF", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>place value</strong> of the following digits from the&nbsp;decimal&nbsp;$\\var{dec}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{tenthsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredthsF}$</td>\n<td style=\"text-align: center;\">$\\var{thousandthsF}$</td>\n</tr>\n<tr>\n<td>Place Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "stepsPenalty": "3", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The&nbsp;<strong>place value</strong> of a digit is the 'value' of the 'column' it is written in. It is all about where the digit is. To the right of the decimal point we have the <em>tenths</em> column, next we have the <em>hundredths</em> column,&nbsp; and then we have the <em>thousandths</em> column. The place value for something in the&nbsp;tenths&nbsp;column is $0.1$, the place value for something in the hundredths column is $0.01$ and the place value for something in the thousandths&nbsp;column is $0.001$.</p>\n<p>For example, the place value of the $3$ in the number $1.234$ is $0.01$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1/b", "maxValue": "1/b", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1/b^2", "maxValue": "1/b^2", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1/b^3", "maxValue": "1/b^3", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Please enter the <strong>actual value</strong> of the following digits from the&nbsp;decimal&nbsp;$\\var{dec}$ in the gaps below.</p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">Digit</td>\n<td style=\"text-align: center;\">$\\var{tenthsF}$</td>\n<td style=\"text-align: center;\">$\\var{hundredthsF}$</td>\n<td style=\"text-align: center;\">$\\var{thousandthsF}$</td>\n</tr>\n<tr>\n<td>Actual Value</td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>", "stepsPenalty": "3", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The&nbsp;<strong>actual value</strong> of a digit is the face value times the place value. It is all about what that digit actually adds/brings to the number, what it is actually worth or what it actually represents.</p>\n<p>For example, the actual value of the digit $3$ in the number $1.234$ is $0.03$ (since the $3$ in $1.234$ actually stands for $0.03$).</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tenthsA", "maxValue": "tenthsA", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "hundredthsA", "maxValue": "hundredthsA", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "thousandthsA", "maxValue": "thousandthsA", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Addition and Subtraction", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Addition ", "Subtraction"], "variable_overrides": [[], []], "questions": [{"name": "Merryn's copy of Addition 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["adding", "Addition", "addition", "algorithms", "plus"], "metadata": {"description": "<p>Natural numbers addition algorithm. 2 and 3 digit numbers. Carrying.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following questions down on paper and evaluate them without using a calculator. You can use any method you wish to practice.</p>\n<p>If you click on <em>Show steps</em> you will see full working for the standard method. Click on <em>Try another question like this one</em> to get a new pair of numbers to add.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"chuncarry": {"name": "chuncarry", "group": "Ungrouped variables", "definition": "floor(chunsum/10)", "description": "", "templateType": "anything", "can_override": false}, "ctencarry": {"name": "ctencarry", "group": "Ungrouped variables", "definition": "floor(ctensum/10)", "description": "", "templateType": "anything", "can_override": false}, "bTenSum": {"name": "bTenSum", "group": "Ungrouped variables", "definition": "bdigs[1]+bdigs[3]+bunitsumtensdigit", "description": "", "templateType": "anything", "can_override": false}, "cunitsum": {"name": "cunitsum", "group": "Ungrouped variables", "definition": "cdigs[0]+cdigs[3]", "description": "", "templateType": "anything", "can_override": false}, "bUnitSumLastDigit": {"name": "bUnitSumLastDigit", "group": "Ungrouped variables", "definition": "mod(bunitsum,10)", "description": "", "templateType": "anything", "can_override": false}, "bunitsumtensdigit": {"name": "bunitsumtensdigit", "group": "Ungrouped variables", "definition": "floor((bunitsum)/10)", "description": "", "templateType": "anything", "can_override": false}, "bans": {"name": "bans", "group": "Ungrouped variables", "definition": "b2digit+b3digit", "description": "", "templateType": "anything", "can_override": false}, "btencarry": {"name": "btencarry", "group": "Ungrouped variables", "definition": "floor(btensum/10)", "description": "", "templateType": "anything", "can_override": false}, "bdigs": {"name": "bdigs", "group": "Ungrouped variables", "definition": "shuffle(1..9)[0..5]", "description": "", "templateType": "anything", "can_override": false}, "cunitcarry": {"name": "cunitcarry", "group": "Ungrouped variables", "definition": "floor(cunitsum/10)", "description": "", "templateType": "anything", "can_override": false}, "bunitsum": {"name": "bunitsum", "group": "Ungrouped variables", "definition": "bdigs[0]+bdigs[2]", "description": "", "templateType": "anything", "can_override": false}, "threedigit2": {"name": "threedigit2", "group": "Ungrouped variables", "definition": "cdigs[3]+cdigs[4]*10+cdigs[5]*100", "description": "", "templateType": "anything", "can_override": false}, "threedigit1": {"name": "threedigit1", "group": "Ungrouped variables", "definition": "cdigs[0]+cdigs[1]*10+cdigs[2]*100", "description": "", "templateType": "anything", "can_override": false}, "chunsumlastdigit": {"name": "chunsumlastdigit", "group": "Ungrouped variables", "definition": "mod(chunsum,10)", "description": "", "templateType": "anything", "can_override": false}, "cdigs": {"name": "cdigs", "group": "Ungrouped variables", "definition": "shuffle(3..9)", "description": "", "templateType": "anything", "can_override": false}, "b2digit": {"name": "b2digit", "group": "Ungrouped variables", "definition": "bdigs[0]+bdigs[1]*10", "description": "", "templateType": "anything", "can_override": false}, "cans": {"name": "cans", "group": "Ungrouped variables", "definition": "threedigit1+threedigit2", "description": "", "templateType": "anything", "can_override": false}, "cunitsumlastdigit": {"name": "cunitsumlastdigit", "group": "Ungrouped variables", "definition": "mod(cunitsum,10)", "description": "", "templateType": "anything", "can_override": false}, "b3digit": {"name": "b3digit", "group": "Ungrouped variables", "definition": "bdigs[2]+bdigs[3]*10+bdigs[4]*100", "description": "", "templateType": "anything", "can_override": false}, "ctensumlastdigit": {"name": "ctensumlastdigit", "group": "Ungrouped variables", "definition": "mod(ctensum,10)", "description": "", "templateType": "anything", "can_override": false}, "bsumhun": {"name": "bsumhun", "group": "Ungrouped variables", "definition": "bdigs[4]+btencarry", "description": "", "templateType": "anything", "can_override": false}, "bsumhunlastdigit": {"name": "bsumhunlastdigit", "group": "Ungrouped variables", "definition": "mod(bsumhun,10)", "description": "", "templateType": "anything", "can_override": false}, "bsumhuncarry": {"name": "bsumhuncarry", "group": "Ungrouped variables", "definition": "floor(bsumhun/10)", "description": "", "templateType": "anything", "can_override": false}, "ctensum": {"name": "ctensum", "group": "Ungrouped variables", "definition": "cdigs[1]+cdigs[4]+cunitcarry", "description": "", "templateType": "anything", "can_override": false}, "chunsum": {"name": "chunsum", "group": "Ungrouped variables", "definition": "ctencarry+cdigs[2]+cdigs[5]", "description": "", "templateType": "anything", "can_override": false}, "btensumlastdigit": {"name": "btensumlastdigit", "group": "Ungrouped variables", "definition": "mod(btensum,10)", "description": "", "templateType": "anything", "can_override": false}, "cunitsumtensdigit": {"name": "cunitsumtensdigit", "group": "Ungrouped variables", "definition": "0.1*(cunitsum-cunitsumlastdigit)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["bdigs", "b2digit", "b3digit", "bunitsum", "bUnitSumLastDigit", "bunitsumtensdigit", "bTenSum", "btensumlastdigit", "btencarry", "bsumhun", "bsumhuncarry", "bsumhunlastdigit", "bans", "cdigs", "threedigit1", "threedigit2", "cans", "cunitsum", "cunitsumlastdigit", "cunitcarry", "ctensum", "ctensumlastdigit", "ctencarry", "chunsum", "chunsumlastdigit", "chuncarry", "cunitsumtensdigit"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{b2digit}+\\var{b3digit} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally, we set up $\\var{b2digit}+\\var{b3digit}$ with the <em>ones</em> and the <em>tens</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td>$\\var{bdigs[1]}$</td>\n<td>$\\var{bdigs[0]}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bdigs[4]}$</td>\n<td>$\\var{bdigs[3]}$</td>\n<td>$\\var{bdigs[2]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we add the digits in the <g class=\"gr_ gr_25 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"25\" data-gr-id=\"25\">ones</g> column.</p>\n<p><span data-jme-visible=\"bunitsumtensdigit=0\">This results in&nbsp;$\\var{bunitsum}$ and so&nbsp;we place $\\var{bunitsumlastdigit}$ under the line in the ones column. </span></p>\n<p><span data-jme-visible=\"bunitsumtensdigit=1\">This results in&nbsp;$\\var{bunitsum}$ and so&nbsp;we place $\\var{bunitsumlastdigit}$ under the line in the <g class=\"gr_ gr_28 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"28\" data-gr-id=\"28\">ones</g> column and carry the $1$ into the tens column.&nbsp;</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"bunitsumtensdigit=1\">$\\overset{\\color{red}1}{\\var{bdigs[1]}}$</span> <span data-jme-visible=\"bunitsumtensdigit=0\">$\\overset{\\phantom{0}}{\\var{bdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{bdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bdigs[4]}$</td>\n<td>$\\var{bdigs[3]}$</td>\n<td>$\\var{bdigs[2]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{bUnitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the tens column.</p>\n<p><span data-jme-visible=\"btencarry=0\">This results in&nbsp;$\\var{bTenSum}$ and so we place $\\var{bTenSumlastdigit}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"btencarry=1\">This results in&nbsp;$\\var{bTenSum}$ and so we place $\\var{bTenSumlastdigit}$ under the line in the tens column and carry the $1$ into the hundreds column.</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"btencarry=1\">${\\overset{\\color{red}1}{\\phantom{0}}}$</span> <span data-jme-visible=\"btencarry=0\">$\\phantom{0}$</span></td>\n<td><span data-jme-visible=\"bunitsumtensdigit=1\">$\\overset{1}{\\var{bdigs[1]}}$</span> <span data-jme-visible=\"bunitsumtensdigit=0\">$\\overset{\\phantom{0}}{\\var{bdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{bdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bdigs[4]}$</td>\n<td>$\\var{bdigs[3]}$</td>\n<td>$\\var{bdigs[2]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{bTenSumlastdigit}}$</td>\n<td>${\\var{bUnitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the hundreds column.</p>\n<p><span data-jme-visible=\"btencarry=0\">This is simply $\\var{bdigs[4]}$ so we place&nbsp;$\\var{bdigs[4]}$ under the line in the hundreds column.</span></p>\n<p><span data-jme-visible=\"btencarry=1\">This is $\\var{bsumhun}$ so we place&nbsp;$\\var{bsumhun}$ under the line in the hundreds column.</span></p>\n<p><span data-jme-visible=\"bans&gt;999\">This is $\\var{bsumhun}$ so we place&nbsp;$\\var{bsumhun}$ under the line so that its rightmost digit is in the hundreds column.</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"btencarry=1\">$\\overset{1}{\\phantom{0}}$</span> <span data-jme-visible=\"btencarry=0\">$\\phantom{0}$</span></td>\n<td><span data-jme-visible=\"bunitsumtensdigit=1\">$\\overset{1}{\\var{bdigs[1]}}$</span> <span data-jme-visible=\"bunitsumtensdigit=0\">$\\overset{\\phantom{0}}{\\var{bdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{bdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bdigs[4]}$</td>\n<td>$\\var{bdigs[3]}$</td>\n<td>$\\var{bdigs[2]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"bsumhuncarry=0\">$\\phantom{\\var{bsumhuncarry}}$</span> <span data-jme-visible=\"bsumhuncarry=1\">$\\color{red}{\\var{bsumhuncarry}}$</span></td>\n<td>$\\color{red}{\\var{bsumhunlastdigit}}$</td>\n<td>$\\var{bTenSumlastdigit}$</td>\n<td>${\\var{bUnitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{bans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "bans", "maxValue": "bans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{threedigit1}+\\var{threedigit2} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally&nbsp;we set up $\\var{threedigit1}+\\var{threedigit2}$&nbsp;with the <em>ones,&nbsp;</em><em>tens</em> and&nbsp;<em>hundreds</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\var{cdigs[2]}$</td>\n<td>$\\var{cdigs[1]}$</td>\n<td>$\\var{cdigs[0]}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we add the digits in the ones column.</p>\n<p><span data-jme-visible=\"cunitsumtensdigit=0\">This results in&nbsp;$\\var{cunitsum}$ and so&nbsp;we place $\\var{cunitsumlastdigit}$ under the line in the ones column. </span></p>\n<p><span data-jme-visible=\"cunitsumtensdigit=1\">This results in&nbsp;$\\var{cunitsum}$ and so&nbsp;we place $\\var{cunitsumlastdigit}$ under the line in the ones column and carry the $1$ into the tens column.&nbsp;</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{\\color{red}1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the tens column.</p>\n<p><span data-jme-visible=\"ctencarry=0\">This results in&nbsp;$\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ctencarry=1\">This results in&nbsp;$\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in the tens column and carry the $1$ into the hundreds column.</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ctencarry=1\">${\\overset{\\color{red}1}{\\var{cdigs[2]}}}$</span> <span data-jme-visible=\"ctencarry=0\">$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</span></td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ctenSumlastdigit}}$</td>\n<td>${\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the hundreds column.</p>\n<p><span data-jme-visible=\"cans&lt;1000\">This is $\\var{chunsum}$ so we place&nbsp;$\\var{chunsum}$ under the line in the hundreds column.</span></p>\n<p><span data-jme-visible=\"cans&gt;999\">This is $\\var{chunsum}$ so we place&nbsp;$\\var{chunsum}$ under the line so that its rightmost digit is in the hundreds column.</span></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ctencarry=1\">$\\overset{1}{\\var{cdigs[2]}}$</span> <span data-jme-visible=\"ctencarry=0\">$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</span></td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"chuncarry=0\">$\\phantom{\\var{chuncarry}}$</span> <span data-jme-visible=\"chuncarry=1\">$\\color{red}{\\var{chuncarry}}$</span></td>\n<td>$\\color{red}{\\var{chunsumlastdigit}}$</td>\n<td>$\\var{ctenSumlastdigit}$</td>\n<td>${\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{cans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "cans", "maxValue": "cans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["algorithms", "minus", "subtracting", "subtraction", "take away"], "metadata": {"description": "<p>The subtraction algortihm using the borrow and pay back method with integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate it without using a calculator using any method you choose.</p>\n<p>If you click on <em>Show steps</em> you will see full working using the standard algorithm. Click on <em>Try another question like this one</em> to get a new pair of numbers.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"newtopten": {"name": "newtopten", "group": "c", "definition": "if(unitdiff>=0,top[1],top[1]-1)", "description": "", "templateType": "anything", "can_override": false}, "ansunit": {"name": "ansunit", "group": "c", "definition": "mod(ans,10)", "description": "", "templateType": "anything", "can_override": false}, "hundiff": {"name": "hundiff", "group": "c", "definition": "if(tendiff>=0,top[2]-bot[2],top[2]-1-bot[2])", "description": "", "templateType": "anything", "can_override": false}, "botnum": {"name": "botnum", "group": "c", "definition": "bot[0]+bot[1]*10+bot[2]*100", "description": "", "templateType": "anything", "can_override": false}, "ansten": {"name": "ansten", "group": "c", "definition": "mod(floor(ans/10),10)", "description": "", "templateType": "anything", "can_override": false}, "tendiff": {"name": "tendiff", "group": "c", "definition": "if(unitdiff>=0,top[1]-bot[1],top[1]-1-bot[1])", "description": "", "templateType": "anything", "can_override": false}, "unitdiff": {"name": "unitdiff", "group": "c", "definition": "top[0]-bot[0]", "description": "", "templateType": "anything", "can_override": false}, "bot": {"name": "bot", "group": "c", "definition": "random(\nif(top[0]<9,[random(top[0]+1..9), random(top[1]..9), random(1..top[2]-1)],\n      if(top[1]<9,[random(top[0]..9), random(top[1]+1..9), random(1..top[2]-1)],\"error\")),\nif(top[1]<9,[random(0..top[0]), random(top[1]+1..9), random(1..top[2]-1)],\n      if(top[1]=9,[random(top[0]..9), random(0..9), random(1..top[2]-1)],\"error\"))\n)\n", "description": "<p>This should force some borrowing and <g class=\"gr_ gr_4 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"4\" data-gr-id=\"4\">paying</g> back, and that the&nbsp;final answer&nbsp;is positive.</p>", "templateType": "anything", "can_override": false}, "newtophun": {"name": "newtophun", "group": "c", "definition": "if(tendiff>=0,top[2],top[2]-1)", "description": "", "templateType": "anything", "can_override": false}, "topnum": {"name": "topnum", "group": "c", "definition": "top[0]+top[1]*10+top[2]*100", "description": "", "templateType": "anything", "can_override": false}, "anshun": {"name": "anshun", "group": "c", "definition": "mod(floor(ans/100),10)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "c", "definition": "topnum-botnum", "description": "", "templateType": "anything", "can_override": false}, "top": {"name": "top", "group": "c", "definition": "random([random(0..9),random(1..8),random(2..9)],[random(0..8),random(1..9),random(2..9)])", "description": "<p>Borrowing from the hundreds to do the units is not covered with this randomisation. We will do that in another part.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "c", "variables": ["top", "bot", "topnum", "botnum", "ans", "unitdiff", "tendiff", "hundiff", "ansunit", "ansten", "anshun", "newtopten", "newtophun"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{topnum}-\\var{botnum} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally&nbsp;we set up $\\var{topnum}-\\var{botnum}$&nbsp;with the <em>ones,&nbsp;</em><em>tens</em> and&nbsp;<em>hundreds</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\var{top[2]}$</td>\n<td>$\\var{top[1]}$</td>\n<td>$\\var{top[0]}$</td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we try to subtract the digits in the <g class=\"gr_ gr_38 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"38\" data-gr-id=\"38\">ones</g> column.</p>\n<p><span data-jme-visible=\"unitdiff&gt;=0\">Since this is $\\var{ansunit}$ we&nbsp;write $\\var{ansunit}$ under the line in the ones column.</span></p>\n<p><span data-jme-visible=\"unitdiff&lt;0\">Since we can't&nbsp;take $\\var{bot[0]}$ away from $\\var{top[0]}$ (without using negative numbers) we <em>borrow</em>&nbsp;a ten from the tens column. This means we cross out the $\\var{top[1]}$ in the tens column&nbsp;and replace it with a $\\var{top[1]-1}$, and the $\\var{top[0]}$ becomes a $\\var{10+top[0]}$. Now we can do&nbsp;$\\var{10+top[0]}-\\var{bot[0]}$, and write the result, $\\var{ansunit}$, under the line in the ones column.</span>&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{top[2]}}$</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\overset{\\color{red}{\\var{newtopten}}}{\\var{top[1]}\\mkern-7.5mu\\color{red}/}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ansunit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the tens column.</p>\n<p><span data-jme-visible=\"tendiff&gt;=0\">Since this is $\\var{ansten}$ we&nbsp;write $\\var{ansten}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"tendiff&lt;0\">Since we can't&nbsp;take $\\var{bot[1]}$ away from $\\var{newtopten}$ (without using negative numbers) we <em>borrow</em>&nbsp;a hundred from the hundred column. This means we cross out the $\\var{top[2]}$ in the hundreds column&nbsp;and replace it with a $\\var{top[2]-1}$, and the $\\var{newtopten}$ in the tens column becomes a $\\var{10+newtopten}$. Now we can do&nbsp;$\\var{10+newtopten}-\\var{bot[1]}$, and write the result, $\\var{ansten}$, under the line in the tens column.</span>&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\color{red}{\\var{newtophun}}}{\\var{top[2]}\\mkern-7.5mu\\color{red}{/}}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[2]}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{\\color{red}{1}\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\var{top[1]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ansten}}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the hundreds column.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\var{newtophun}}{\\var{top[2]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[2]}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{{1}\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[1]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{anshun}}$</td>\n<td>$\\var{ansten}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>The answer is therefore $\\var{ans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Multiplication", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Multiplication 2 digit by 2 digit", "Multiplication 3 digit by 2 digit", "Multiplication as division"], "variable_overrides": [[], [], []], "questions": [{"name": "Merryn's copy of Multiplication 2 digit by 2 digit", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["algorithms", "long multiplication", "multiplication", "times"], "metadata": {"description": "<p>Multiplication algorithm with integers</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate it without using a calculator using whichever method you choose.&nbsp;</p>\n<p>If you click on <em>Show steps</em> you will see the full working for the standard method. Click on <em>Try another question like this one</em> to get new numbers to practise with.&nbsp;</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ab0t0": {"name": "ab0t0", "group": "2digit", "definition": "atop[0]*abot[0]", "description": "", "templateType": "anything", "can_override": false}, "asum1": {"name": "asum1", "group": "2digit", "definition": "abot[0]*atopnum", "description": "", "templateType": "anything", "can_override": false}, "ab0t1": {"name": "ab0t1", "group": "2digit", "definition": "abot[0]*atop[1]", "description": "", "templateType": "anything", "can_override": false}, "ab0t0carry": {"name": "ab0t0carry", "group": "2digit", "definition": "(ab0t0-ab0t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "abot": {"name": "abot", "group": "2digit", "definition": "if(adigs[2]<>0,[adigs[3],adigs[2]],[adigs[2],adigs[3]])", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">abot</g></p>", "templateType": "anything", "can_override": false}, "abotnum": {"name": "abotnum", "group": "2digit", "definition": "abot[1]*10+abot[0]", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">botnum</g></p>", "templateType": "anything", "can_override": false}, "aansone": {"name": "aansone", "group": "2digit", "definition": "mod(aans,10)", "description": "", "templateType": "anything", "can_override": false}, "aansten": {"name": "aansten", "group": "2digit", "definition": "mod((aans-aansone)/10,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t0last": {"name": "ab1t0last", "group": "2digit", "definition": "mod(ab1t0,10)", "description": "", "templateType": "anything", "can_override": false}, "threedigit2": {"name": "threedigit2", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "twodigit3": {"name": "twodigit3", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "aans": {"name": "aans", "group": "2digit", "definition": "atopnum*abotnum", "description": "", "templateType": "anything", "can_override": false}, "ab1t1last": {"name": "ab1t1last", "group": "2digit", "definition": "mod(ab1t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t1carry": {"name": "ab1t1carry", "group": "2digit", "definition": "(ab1t1pluscarry-ab1t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "aanstho": {"name": "aanstho", "group": "2digit", "definition": "mod((aans-aansone-aansten*10-aanshun*100)/1000,10)", "description": "", "templateType": "anything", "can_override": false}, "aanshun": {"name": "aanshun", "group": "2digit", "definition": "mod((aans-aansone-aansten*10)/100,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t0": {"name": "ab1t0", "group": "2digit", "definition": "abot[1]*atop[0]", "description": "", "templateType": "anything", "can_override": false}, "twodigit1": {"name": "twodigit1", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "atopnum": {"name": "atopnum", "group": "2digit", "definition": "atop[1]*10+atop[0]", "description": "", "templateType": "anything", "can_override": false}, "atop": {"name": "atop", "group": "2digit", "definition": "if(adigs[0]<>0,[adigs[1],adigs[0]],[adigs[0],adigs[1]])", "description": "", "templateType": "anything", "can_override": false}, "ab1t1pluscarry": {"name": "ab1t1pluscarry", "group": "2digit", "definition": "ab1t1+ab1t0carry", "description": "", "templateType": "anything", "can_override": false}, "ab0t1carry": {"name": "ab0t1carry", "group": "2digit", "definition": "(ab0t1pluscarry-ab0t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "threedigit3": {"name": "threedigit3", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "twodigit2": {"name": "twodigit2", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "asum2": {"name": "asum2", "group": "2digit", "definition": "10*abot[1]*atopnum", "description": "<p>sum2</p>", "templateType": "anything", "can_override": false}, "threedigit1": {"name": "threedigit1", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "ab0t1pluscarry": {"name": "ab0t1pluscarry", "group": "2digit", "definition": "ab0t1+ab0t0carry", "description": "<p>ab</p>", "templateType": "anything", "can_override": false}, "ab1t0carry": {"name": "ab1t0carry", "group": "2digit", "definition": "(ab1t0-ab1t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "ab0t1last": {"name": "ab0t1last", "group": "2digit", "definition": "mod(ab0t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t0last": {"name": "ab0t0last", "group": "2digit", "definition": "mod(ab0t0,10)", "description": "", "templateType": "anything", "can_override": false}, "adigs": {"name": "adigs", "group": "2digit", "definition": "shuffle(1..9)[0..4]", "description": "<p>we want distinct digits so it is easier to refer to digits unambiguously.&nbsp;</p>", "templateType": "anything", "can_override": false}, "ab1t1": {"name": "ab1t1", "group": "2digit", "definition": "abot[1]*atop[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["twodigit1", "twodigit2", "threedigit1", "threedigit2", "twodigit3", "threedigit3"], "variable_groups": [{"name": "2digit", "variables": ["adigs", "atop", "abot", "atopnum", "abotnum", "aans", "ab0t0", "ab0t0last", "ab0t0carry", "ab0t1", "ab0t1pluscarry", "ab0t1last", "ab0t1carry", "ab1t0", "ab1t0last", "ab1t0carry", "ab1t1", "ab1t1pluscarry", "ab1t1last", "ab1t1carry", "asum1", "asum2", "aansone", "aansten", "aanshun", "aanstho"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{atopnum}\\times\\var{abotnum} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally&nbsp;we set up $\\var{atopnum}\\times\\var{abotnum}$&nbsp;with the <em>ones </em>and<em>&nbsp;</em><em>tens</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\var{atop[1]}$</td>\n<td>$\\var{atop[0]}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>We need to multiply each digit in the bottom number by each digit in the top number whilst respecting their place values.</p>\n<p>&nbsp;</p>\n<p><strong>We multiply the digits in the ones column, that is, $\\color{green}{\\var{abot[0]}\\times \\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab0t0&lt;10\">Since this is just $\\var{ab0t0}$ we&nbsp;write $\\var{ab0t0}$ under the line in the ones column.</span></p>\n<p><span data-jme-visible=\"ab0t0&gt;=10\">Since this is $\\var{ab0t0}$ we write the $\\var{ab0t0last}$ under the line in the ones column and carry the&nbsp;$\\var{ab0t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab0t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p><strong>We now multiply&nbsp;diagonally, $\\color{green}{\\var{abot[0]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab0t1&lt;10 and ab0t0carry=0\">This just gives us $\\var{ab0t1}$ so we write&nbsp;$\\var{ab0t1}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t1&gt;=10 and ab0t0carry=0\">This gives us $\\var{ab0t1}$ so we write this under the line with the $\\var{ab0t1last}$ in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&gt;=10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1pluscarry}$ under the line with the $\\var{ab0t1last}$ in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&lt;10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1pluscarry}$ under the line in the tens column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;&nbsp;</td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">$\\color{green}{\\overset{{\\var{ab0t0carry}}}{\\var{atop[1]}}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>&nbsp;</td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\color{red}{\\var{ab0t1carry}}$</span> <span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td>&nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>We are now finished with the digit $\\var{abot[0]}$ and move on to work with the $\\var{abot[1]}$ in the tens column. Since this is really a $\\var{abot[1]*10}$ <strong>we place a zero in the ones column on the next line</strong> to pad our numbers out. <span data-jme-visible=\"ab0t0carry&lt;&gt;0\">We also crossout or erase any carry marks that we have used.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{atop[1]}}$</td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}0$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply along the other diagonal, that is, $\\color{green}{\\var{abot[1]}\\times\\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab1t0&lt;10\">Since this is just $\\var{ab1t0}$ we&nbsp;write $\\var{ab1t0}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab1t0&gt;=10\">Since this is $\\var{ab1t0}$ we write the $\\var{ab1t0last}$ under the line in the tens column and carry the&nbsp;$\\var{ab1t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply the digits in the tens column, that is, $\\color{green}{\\var{abot[1]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab1t1&lt;10 and ab1t0carry=0\">This just gives us $\\var{ab1t1}$ so we write&nbsp;$\\var{ab1t1}$ under the line in the&nbsp;hundreds column.</span></p>\n<p><span data-jme-visible=\"ab1t1&gt;=10 and ab1t0carry=0\">This gives us $\\var{ab1t1}$ so we write this under the line with the $\\var{ab1t1last}$ in the&nbsp;hundreds&nbsp;column.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&gt;=10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1pluscarry}$ under the line with the $\\var{ab1t1last}$ in the hundreds column.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&lt;10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1pluscarry}$ under the line in the&nbsp;hundreds column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\color{green}{\\overset{{\\var{ab1t0carry}}}{\\var{atop[1]}}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\color{red}{\\var{ab1t1carry}}$</span> <span data-jme-visible=\"ab1t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab1t1last}}$</td>\n<td>${\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now add the two results to get the total, that is, $\\color{green}{\\var{asum1}+\\var{asum2}}$.</strong></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\color{green}{\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab0t1last}}$</td>\n<td>$\\color{green}{\\var{ab0t0last}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\color{green}{\\var{ab1t1carry}}$</span> <span data-jme-visible=\"ab1t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab1t1last}}$</td>\n<td>$\\color{green}{\\var{ab1t0last}}$</td>\n<td>$\\color{green}{0}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"aanstho&lt;&gt;0\">$\\color{red}{\\var{aanstho}}$</span><span data-jme-visible=\"aanstho=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{aanshun}}$</td>\n<td>$\\color{red}{\\var{aansten}}$</td>\n<td>$\\color{red}{\\var{aansone}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{aans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "aans", "maxValue": "aans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Multiplication 3 digit by 2 digit", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Multiplication algorithm with integers</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate it without using a calculator using any method you choose.</p>\n<p>If you click on <em>Show steps</em> you will see the full working using the standard method. Click on <em>Try another question like this one</em> to practise with different numbers.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ab0t0last": {"name": "ab0t0last", "group": "2digit", "definition": "mod(ab0t0,10)", "description": "", "templateType": "anything", "can_override": false}, "threedigit3": {"name": "threedigit3", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "aanstho": {"name": "aanstho", "group": "2digit", "definition": "mod((aans-aansone-aansten*10-aanshun*100)/1000,10)", "description": "", "templateType": "anything", "can_override": false}, "aanstth": {"name": "aanstth", "group": "2digit", "definition": "mod((aans-aansone-aansten*10-aanshun*100-aanstho*1000)/10000,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t1": {"name": "ab0t1", "group": "2digit", "definition": "abot[0]*atop[1]", "description": "", "templateType": "anything", "can_override": false}, "threedigit2": {"name": "threedigit2", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "ab1t2pluscarry": {"name": "ab1t2pluscarry", "group": "2digit", "definition": "ab1t2+ab1t1carry", "description": "", "templateType": "anything", "can_override": false}, "twodigit1": {"name": "twodigit1", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "asum1": {"name": "asum1", "group": "2digit", "definition": "abot[0]*atopnum", "description": "", "templateType": "anything", "can_override": false}, "aanshun": {"name": "aanshun", "group": "2digit", "definition": "mod((aans-aansone-aansten*10)/100,10)", "description": "", "templateType": "anything", "can_override": false}, "abot": {"name": "abot", "group": "2digit", "definition": "[adigs[4],adigs[3]]", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">abot</g></p>", "templateType": "anything", "can_override": false}, "threedigit1": {"name": "threedigit1", "group": "Ungrouped variables", "definition": "random(101..999 except list(200..900#100) )", "description": "", "templateType": "anything", "can_override": false}, "ab1t1last": {"name": "ab1t1last", "group": "2digit", "definition": "mod(ab1t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "twodigit3": {"name": "twodigit3", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "adigs": {"name": "adigs", "group": "2digit", "definition": "shuffle(1..9)[0..5]", "description": "<p>we want distinct digits so it is easier to refer to digits unambiguously.&nbsp;</p>", "templateType": "anything", "can_override": false}, "ab1t1": {"name": "ab1t1", "group": "2digit", "definition": "abot[1]*atop[1]", "description": "", "templateType": "anything", "can_override": false}, "ab1t2carry": {"name": "ab1t2carry", "group": "2digit", "definition": "(ab1t2pluscarry-ab1t2last)/10", "description": "", "templateType": "anything", "can_override": false}, "asum2": {"name": "asum2", "group": "2digit", "definition": "10*abot[1]*atopnum", "description": "<p>sum2</p>", "templateType": "anything", "can_override": false}, "atop": {"name": "atop", "group": "2digit", "definition": "[adigs[2],adigs[1],adigs[0]]", "description": "", "templateType": "anything", "can_override": false}, "ab1t0last": {"name": "ab1t0last", "group": "2digit", "definition": "mod(ab1t0,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t2last": {"name": "ab0t2last", "group": "2digit", "definition": "mod(ab0t2pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t1pluscarry": {"name": "ab1t1pluscarry", "group": "2digit", "definition": "ab1t1+ab1t0carry", "description": "", "templateType": "anything", "can_override": false}, "ab1t0": {"name": "ab1t0", "group": "2digit", "definition": "abot[1]*atop[0]", "description": "", "templateType": "anything", "can_override": false}, "ab1t0carry": {"name": "ab1t0carry", "group": "2digit", "definition": "(ab1t0-ab1t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "ab0t1last": {"name": "ab0t1last", "group": "2digit", "definition": "mod(ab0t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t2last": {"name": "ab1t2last", "group": "2digit", "definition": "mod(ab1t2pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t2": {"name": "ab1t2", "group": "2digit", "definition": "abot[1]*atop[2]", "description": "", "templateType": "anything", "can_override": false}, "ab0t2pluscarry": {"name": "ab0t2pluscarry", "group": "2digit", "definition": "ab0t2+ab0t1carry", "description": "", "templateType": "anything", "can_override": false}, "ab0t2carry": {"name": "ab0t2carry", "group": "2digit", "definition": "(ab0t2pluscarry-ab0t2last)/10", "description": "", "templateType": "anything", "can_override": false}, "twodigit2": {"name": "twodigit2", "group": "Ungrouped variables", "definition": "random(11..99 except list(20..90#10) )", "description": "", "templateType": "anything", "can_override": false}, "ab0t0carry": {"name": "ab0t0carry", "group": "2digit", "definition": "(ab0t0-ab0t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "ab1t1carry": {"name": "ab1t1carry", "group": "2digit", "definition": "(ab1t1pluscarry-ab1t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "ab0t2": {"name": "ab0t2", "group": "2digit", "definition": "abot[0]*atop[2]", "description": "", "templateType": "anything", "can_override": false}, "ab0t1carry": {"name": "ab0t1carry", "group": "2digit", "definition": "(ab0t1pluscarry-ab0t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "abotnum": {"name": "abotnum", "group": "2digit", "definition": "abot[1]*10+abot[0]", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">botnum</g></p>", "templateType": "anything", "can_override": false}, "ab0t1pluscarry": {"name": "ab0t1pluscarry", "group": "2digit", "definition": "ab0t1+ab0t0carry", "description": "<p>ab</p>", "templateType": "anything", "can_override": false}, "atopnum": {"name": "atopnum", "group": "2digit", "definition": "atop[2]*100+atop[1]*10+atop[0]", "description": "", "templateType": "anything", "can_override": false}, "aans": {"name": "aans", "group": "2digit", "definition": "atopnum*abotnum", "description": "", "templateType": "anything", "can_override": false}, "aansone": {"name": "aansone", "group": "2digit", "definition": "mod(aans,10)", "description": "", "templateType": "anything", "can_override": false}, "aansten": {"name": "aansten", "group": "2digit", "definition": "mod((aans-aansone)/10,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t0": {"name": "ab0t0", "group": "2digit", "definition": "atop[0]*abot[0]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["twodigit1", "twodigit2", "threedigit1", "threedigit2", "twodigit3", "threedigit3"], "variable_groups": [{"name": "2digit", "variables": ["adigs", "atop", "abot", "atopnum", "abotnum", "aans", "ab0t0", "ab0t0last", "ab0t0carry", "ab0t1", "ab0t1pluscarry", "ab0t1last", "ab0t1carry", "ab0t2", "ab0t2pluscarry", "ab0t2last", "ab0t2carry", "ab1t0", "ab1t0last", "ab1t0carry", "ab1t1", "ab1t1pluscarry", "ab1t1last", "ab1t1carry", "ab1t2", "ab1t2pluscarry", "ab1t2last", "ab1t2carry", "asum1", "asum2", "aansone", "aansten", "aanshun", "aanstho", "aanstth"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{atopnum}\\times\\var{abotnum} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally&nbsp;we set up $\\var{atopnum}\\times\\var{abotnum}$&nbsp;with the <em>ones </em>and<em>&nbsp;</em><em>tens</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\var{atop[2]}$</td>\n<td>$\\var{atop[1]}$</td>\n<td>$\\var{atop[0]}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>We need to multiply each digit in the bottom number by each digit in the top number whilst respecting their place values.</p>\n<p></p>\n<p><strong>We multiply the digits in the ones column, that is, $\\color{green}{\\var{abot[0]}\\times \\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab0t0&lt;10\">Since this is just $\\var{ab0t0}$ we&nbsp;write $\\var{ab0t0}$ under the line in the ones column.</span></p>\n<p><span data-jme-visible=\"ab0t0&gt;=10\">Since this is $\\var{ab0t0}$ we write the $\\var{ab0t0last}$ under the line in the ones column and carry the&nbsp;$\\var{ab0t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{\\var{atop[2]}}$</td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab0t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p><strong>We now multiply&nbsp;diagonally, $\\color{green}{\\var{abot[0]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab0t1&lt;10 and ab0t0carry=0\">This just gives us $\\var{ab0t1}$ so we write&nbsp;$\\var{ab0t1}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t1&gt;=10 and ab0t0carry=0\">This gives us $\\var{ab0t1}$ so we write this under the line with the $\\var{ab0t1last}$ in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&gt;=10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1last}$ under the line in the tens column and carry the $\\var{ab0t1carry}$ into the hundreds column to be dealt with in the next step.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&lt;10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1pluscarry}$ under the line in the tens column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab0t1carry}}}{\\var{atop[2]}}$</span> <span data-jme-visible=\"ab0t1carry=0\">$\\overset{\\phantom{1}}{\\var{atop[2]}}$</span></td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">${\\overset{{\\var{ab0t0carry}}}{\\color{green}{\\var{atop[1]}}}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply&nbsp;diagonally, $\\color{green}{\\var{abot[0]}\\times \\var{atop[2]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab0t2&lt;10 and ab0t1carry=0\">This just gives us $\\var{ab0t2}$ so we write&nbsp;$\\var{ab0t2}$ under the line in the&nbsp;hundreds&nbsp;column.</span></p>\n<p><span data-jme-visible=\"ab0t2&gt;=10 and ab0t1carry=0\">This gives us $\\var{ab0t2}$ so we write this under the line with the $\\var{ab0t2last}$ in the&nbsp;hundreds&nbsp;column.</span></p>\n<p><span data-jme-visible=\"ab0t1carry&lt;&gt;0 and ab0t2pluscarry&gt;=10\">This gives us $\\var{ab0t2}$ but we have to add the $\\var{ab0t1carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t2pluscarry}$ under the line with the $\\var{ab0t2last}$ in the&nbsp;hundreds column.</span></p>\n<p><span data-jme-visible=\"ab0t1carry&lt;&gt;0 and ab0t2pluscarry&lt;10\">This gives us $\\var{ab0t2}$ but we have to add the $\\var{ab0t1carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t2pluscarry}$ under the line in the&nbsp;hundreds column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\overset{{\\var{ab0t1carry}}}{\\color{green}{\\var{atop[2]}}}$</span> <span data-jme-visible=\"ab0t1carry=0\">$\\overset{\\phantom{1}}{\\color{green}{\\var{atop[2]}}}$</span></td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">${\\overset{{\\var{ab0t0carry}}}{\\var{atop[1]}}}$</span> <span data-jme-visible=\"ab0t0carry=0\">${\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t2carry&lt;&gt;0\">$\\color{red}{\\var{ab0t2carry}}$</span><span data-jme-visible=\"ab0t2carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab0t2last}}$</td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>We are now finished with the digit $\\var{abot[0]}$ and move on to work with the $\\var{abot[1]}$ in the tens column. Since this is really a $\\var{abot[1]*10}$ <strong>we place a zero in the ones column on the next line</strong> to pad our numbers out. <span data-jme-visible=\"ab0t0carry&lt;&gt;0\">We also cross out or erase any carry marks that we have used.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{atop[2]}}$</td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[1]}}$</td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td>&nbsp;<span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\var{ab0t2carry}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t2last}}$</td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}0$</td>\n<td>&nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply along the other diagonal, that is, $\\color{green}{\\var{abot[1]}\\times\\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab1t0&lt;10\">Since this is just $\\var{ab1t0}$ we&nbsp;write $\\var{ab1t0}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab1t0&gt;=10\">Since this is $\\var{ab1t0}$ we write the $\\var{ab1t0last}$ under the line in the tens column and carry the&nbsp;$\\var{ab1t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{atop[2]}}$</td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t2carry&lt;&gt;0\">$\\var{ab0t2carry}$</span><span data-jme-visible=\"ab0t2carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t2last}}$</td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p><strong>We now multiply the digits in the tens column, that is, $\\color{green}{\\var{abot[1]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab1t1&lt;10 and ab1t0carry=0\">This just gives us $\\var{ab1t1}$ so we write&nbsp;$\\var{ab1t1}$ under the line in the&nbsp;hundreds column.</span></p>\n<p><span data-jme-visible=\"ab1t1&gt;=10 and ab1t0carry=0\">This gives us $\\var{ab1t1}$ so we write this under the line with the $\\var{ab1t1last}$ in the&nbsp;hundreds&nbsp;column.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&gt;=10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1last}$ under the line in the hundreds column and carry the $\\var{ab1t1carry}$ into the hundreds&nbsp;column to be dealt with in the next step.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&lt;10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1pluscarry}$ under the line in the&nbsp;hundreds column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>&nbsp;<span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab1t1carry}}}{\\var{atop[2]}}$</span>&nbsp;<span data-jme-visible=\"ab1t1carry=0\">$\\overset{\\phantom{1}}{\\var{atop[2]}}$</span></td>\n<td>&nbsp;<span data-jme-visible=\"ab1t0carry&lt;&gt;0\">${\\overset{{\\var{ab1t0carry}}}{\\color{green}{\\var{atop[1]}}}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t2carry&lt;&gt;0\">${\\var{ab0t2carry}}$</span><span data-jme-visible=\"ab0t2carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t2last}}$</td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{ab1t1last}}$</td>\n<td>${\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td>&nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong>We now multiply the final pair of digits, that is, $\\color{green}{\\var{abot[1]}\\times \\var{atop[2]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab1t2&lt;10 and ab1t1carry=0\">This just gives us $\\var{ab1t2}$ so we write&nbsp;$\\var{ab1t2}$ under the line in the&nbsp;thousands column.</span></p>\n<p><span data-jme-visible=\"ab1t2&gt;=10 and ab1t1carry=0\">This gives us $\\var{ab1t2}$ so we write this under the line with the $\\var{ab1t2last}$ in the&nbsp;thousands column.</span></p>\n<p><span data-jme-visible=\"ab1t1carry&lt;&gt;0 and ab1t2pluscarry&gt;=10\">This gives us $\\var{ab1t2}$ but we have to add the $\\var{ab1t1carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t2pluscarry}$ under the line with the $\\var{ab1t2last}$ in the&nbsp;thousands column.</span></p>\n<p><span data-jme-visible=\"ab1t1carry&lt;&gt;0 and ab1t2pluscarry&lt;10\">This gives us $\\var{ab1t2}$ but we have to add the $\\var{ab1t1carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t2pluscarry}$ under the line in the&nbsp;thousands column.</span></p>\n<p></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\overset{{\\var{ab1t1carry}}}{\\color{green}{\\var{atop[2]}}}$</span>&nbsp;<span data-jme-visible=\"ab1t1carry=0\">$\\overset{\\phantom{1}}{\\color{green}{\\var{atop[2]}}}$</span></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">${\\overset{{\\var{ab1t0carry}}}{{\\var{atop[1]}}}}$</span> <span data-jme-visible=\"ab1t0carry=0\">${\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td>&nbsp;&nbsp;</td>\n<td><span data-jme-visible=\"ab0t2carry&lt;&gt;0\">$\\var{ab0t2carry}$</span><span data-jme-visible=\"ab0t2carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t2last}}$</td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td>&nbsp;&nbsp;</td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab1t2carry&lt;&gt;0\">$\\color{red}{\\var{ab1t2carry}}$</span><span data-jme-visible=\"ab1t2carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab1t2last}}$</td>\n<td>${\\var{ab1t1last}}$</td>\n<td>${\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td>&nbsp; &nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p><strong>We now add the two results to get the total, that is, $\\color{green}{\\var{asum1}+\\var{asum2}}$.</strong></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\overset{{\\var{ab1t1carry}}}{\\var{atop[2]}}$ </span> <span data-jme-visible=\"ab1t1carry=0\"> $\\overset{\\phantom{1}}{\\var{atop[2]}}$ </span></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td>&nbsp;</td>\n<td><span data-jme-visible=\"ab0t2carry&lt;&gt;0\">$\\color{green}{\\var{ab0t2carry}}$</span><span data-jme-visible=\"ab0t2carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab0t2last}}$</td>\n<td>$\\color{green}{\\var{ab0t1last}}$</td>\n<td>$\\color{green}{\\var{ab0t0last}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td><span data-jme-visible=\"ab1t2carry&lt;&gt;0\">$\\color{green}{\\var{ab1t2carry}}$</span><span data-jme-visible=\"ab1t2carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab1t2last}}$</td>\n<td>$\\color{green}{\\var{ab1t1last}}$</td>\n<td>$\\color{green}{\\var{ab1t0last}}$</td>\n<td>$\\color{green}{0}$</td>\n<td>&nbsp;&nbsp;</td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"aanstth&lt;&gt;0\">$\\color{red}{\\var{aanstth}}$</span> <span data-jme-visible=\"aanstth=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{aanstho}}$</td>\n<td>$\\color{red}{\\var{aanshun}}$</td>\n<td>$\\color{red}{\\var{aansten}}$</td>\n<td>$\\color{red}{\\var{aansone}}$</td>\n<td>&nbsp;&nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{aans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "aans", "maxValue": "aans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Multiplication as division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Students seem to miss the fact that division is actually multiplication by the reciprocal or the inverse of multiplication. This question attempts to address that.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The <strong>multiplication</strong>&nbsp;button on Sally's scientific calculator is broken. However, all the other buttons continue to work. Sally believes that it is now <strong>impossible</strong> to&nbsp;use her calculator to determine the value of such things as $\\var{num}\\times \\var{den}$.</p>", "advice": "<p>Sally can use her calculator to determine $\\var{num}\\times \\var{den}$ without using the multiplication button since $\\var{num}\\times \\var{den}=\\var{num}\\div\\frac{1}{\\var{den}}$.</p>\n<p></p>\n<p>For example, multiplying a number by 2 is the same as dividing that number by a half.&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"den": {"name": "den", "group": "Ungrouped variables", "definition": "random(2..20)", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "random(20..180)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num", "den"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>&nbsp;Is Sally correct?</p>\n<p></p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>Yes, she is correct.</p>", "<p>No, she is incorrect.</p>"], "matrix": [0, "1"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Division", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Division as multiplication", "Division, single digit divisor results in no remainder - long or short division", "Division, single digit divisor results in a remainder which is expressed as a fraction - long or short division", "Division, single digit divisor results in a remainder, answer expressed as a terminating decimal - long or short division", "Division, single digit divisor results in a remainder, answer expressed as a decimal rounded to 1 decimal place - long or short division", " Division, two digit divisor results in no remainder - long or short division", "Division, two digit divisor results in a remainder which is expressed as a fraction - short or long division", "Division, two digit divisor results in a remainder, answer expressed as a terminating decimal - long or short division", "Division, two digit divisor results in a remainder, answer expressed as a decimal rounded to 1 decimal place - long or short division"], "variable_overrides": [[], [], [], [], [], [], [], [], []], "questions": [{"name": "Merryn's copy of Division as multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Students seem to miss the fact that division is actually multiplication by the reciprocal or the inverse of multiplication. This question attempts to address that.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The expression&nbsp;$\\var{num}\\div \\var{den}$ is one number divided by another. We can get the same answer by doing $\\var{num}$ multiplied by what number?</p>\n<p></p>", "advice": "<p>$\\var{num}\\div \\var{den}=\\var{num}\\times\\frac{1}{\\var{den}}$.</p>\n<p></p>\n<p>For example, dividing a number by 2 is the same as multiplying that number by a half.&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"num": {"name": "num", "group": "Ungrouped variables", "definition": "random(20..180)", "description": "", "templateType": "anything", "can_override": false}, "den": {"name": "den", "group": "Ungrouped variables", "definition": "random(2..20)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num", "den"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\frac{1}{\\var{den}}$", "$\\var{den}$"], "matrix": ["1", "0"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, single digit divisor results in no remainder - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>The simplest case. Divisor is single digit. There is no remainder.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/100),10)", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1,10)", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..9 except 10)", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(quotient1,10)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/100),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#100))", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the thousands column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the hundreds column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the thousands column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the hundreds column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}\\,{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the hundreds column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tens column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the hundreds column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tens column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the ones column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We made it to exactly $\\var{b0}$, so there is no remainder in the ones column and no remainder at the end of the process!</p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^{\\var{diff1}}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p>Now we have run out of digits and we have no remainder. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p></p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the thousands column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{555}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the thousands column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{555}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the hundreds column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the&nbsp;hundreds column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundreds column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tens column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tens&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;ones column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><strong></strong></p>\n<p></p>\n<p>Now we have run out of digits and we have no remainder. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, single digit divisor results in a remainder which is expressed as a fraction - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is single digit. There is&nbsp;a remainder which we express as a fraction.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1+remainder", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "", "templateType": "anything", "can_override": false}, "gcd": {"name": "gcd", "group": "Ungrouped variables", "definition": "gcd(remainder,divisor1)", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..9 except 10)", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/100),10)", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#100))", "description": "<p>this is the quotient without reference to the remainder (so we don't get floating point errors)</p>", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "random(1..divisor1-1)", "description": "", "templateType": "anything", "can_override": false}, "newden": {"name": "newden", "group": "Ungrouped variables", "definition": "divisor1/gcd", "description": "", "templateType": "anything", "can_override": false}, "newnum": {"name": "newnum", "group": "Ungrouped variables", "definition": "remainder/gcd", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/100),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1,10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "gcd", "newnum", "newden"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]] and&nbsp;[[1]]</p>\n<p></p>\n<p><strong>Note</strong>, your answer should involve a whole number and a fraction. Put the whole number in the first field and the fraction in the second. For example, if your answer was $4\\frac{3}{5}$ put $4$ in the first field and $3/5$ in the second field.</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the thousands column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the hundreds column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the thousands column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the hundreds column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}\\,{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the hundreds column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tens column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the hundreds column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tens column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the ones column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We only made it to $\\var{prod0}$ but we were aiming for $\\var{b0}$, so we were $\\var{diff0}$ away. This is the <em>remainder</em> (what remains to be divided).&nbsp;</p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^{\\var{diff1}}{\\var{dd0}}}\\end{array} \\begin{array}{r} \\quad \\color{red}{\\text{r}\\,\\var{diff0}}\\\\\\,\\end{array}$</p>\n<p></p>\n<p>Now we have run out of digits and we are left with&nbsp;a remainder of $\\var{remainder}$. Note this&nbsp;$\\var{remainder}$ hasn't been divided by $\\var{divisor1}$ yet (that's what remainder means!). Recall that dividing $\\var{remainder}$ by&nbsp;$\\var{divisor1}$ is the same as the fraction $\\frac{\\var{remainder}}{\\var{divisor1}}$<span data-jme-visible=\"gcd&lt;&gt;1\">&nbsp;which by removing the common factor of $\\var{gcd}$ from the top and the bottom is equivalent to the fraction $\\frac{\\var{newnum}}{\\var{newden}}$</span>.</p>\n<p></p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}+\\frac{\\var{newnum}}{\\var{newden}}$, which we&nbsp;usually write without the plus sign as the mixed number $\\var{quotient1}\\frac{\\var{newnum}}{\\var{newden}}$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p></p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the thousands column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{555}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the thousands column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{555}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the hundreds column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the&nbsp;hundreds column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundreds column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tens column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tens&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;ones column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we have run out of digits and we are left with&nbsp;a remainder of $\\var{remainder}$. Note this&nbsp;$\\var{remainder}$ hasn't been divided by $\\var{divisor1}$ yet (that's what remainder means!). Recall that dividing $\\var{remainder}$ by&nbsp;$\\var{divisor1}$ is the same as the fraction $\\frac{\\var{remainder}}{\\var{divisor1}}$<span data-jme-visible=\"gcd&lt;&gt;1\">&nbsp;which by removing the common factor of $\\var{gcd}$ from the top and the bottom is equivalent to the fraction $\\frac{\\var{newnum}}{\\var{newden}}$</span>.</p>\n<p></p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}+\\frac{\\var{newnum}}{\\var{newden}}$, which we&nbsp;usually write without the plus sign as the mixed number $\\var{quotient1}\\frac{\\var{newnum}}{\\var{newden}}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{remainder}/{divisor1}", "maxValue": "{remainder}/{divisor1}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, single digit divisor results in a remainder, answer expressed as a terminating decimal - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is single digit. There is a remainder which we express as a decimal by continuing the division process. No rounding is required by design (another question will include rounding off).</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1+remainder", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..9 except 5)", "description": "<p>excluded 5 so that the decimal part is longer than 1 place.</p>", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "0\n//random(1..divisor1-1)", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1*100,10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]&nbsp;</p>\n<p></p>\n<p><strong>Note</strong>, your answer should be a decimal.</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this $\\var{dd3}$ actually represents $\\var{dd3*10}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}}.\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}\\,.\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the ones column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,.\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,.\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\"</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,.\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tenths column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tenths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.1}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the hundredths column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the hundredths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundredths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.01}$)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We made it exactly to $\\var{b0}$ so there is no remainder.</p>\n<p></p>\n<p>Now we have run out of digits and we are left with no remainder.&nbsp;</p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>\n<p></p>\n<p></p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>&nbsp;</p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;tens column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*10}$ since it is in the tens column)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the&nbsp;tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the&nbsp;tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the&nbsp;tens column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ does actually represent $\\var{b2}$ since it is in the ones column)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the&nbsp;ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the ones&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tenths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\var{diff2}\\phantom{.}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1/10}$ since it is in the tenths column)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tenths&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tenths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;hundredths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;hundredths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;(note this&nbsp;$\\var{b0}$ actually represents&nbsp;$\\var{b0/100}$ since it is in the hundredths column)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths&nbsp;column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;hundredths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundredths&nbsp;column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we have run out of digits and we are left with no remainder.&nbsp;</p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, single digit divisor results in a remainder, answer expressed as a decimal rounded to 1 decimal place - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is single digit. 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Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..9 except 5)", "description": "<p>excluded 5 so that the decimal part is longer than 1 place.</p>", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1*100,10)", "description": "", "templateType": "anything", "can_override": false}, "prod3": 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"qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(quotient1,1)", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]&nbsp;(1 decimal place)</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>Actually, since we want the answer to <strong>one decimal place</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have <strong>two decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}}$</p>\n<p>Why two? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this $\\var{dd3}$ actually represents $\\var{dd3*10}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}}.\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}\\,.\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the ones column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,.\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,.\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\"</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,.\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tenths column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tenths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.1}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the hundredths column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the hundredths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundredths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.01}$)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p><strong>Step 2:&nbsp;</strong>We could work out the remainder, we could keep adding zeros and continue the procedure but we only needed&nbsp;to determine&nbsp;the second decimal place in order to correctly round to one decimal place and so we now stop the procedure.</p>\n<p>Since the second decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span><span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$.&nbsp;Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (1 dec. pl.).</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>Actually, since we want the answer to <strong>one decimal place</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have <strong>two decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}}$</p>\n<p>Why two? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;tens column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*10}$ since it is in the tens column)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the&nbsp;tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the&nbsp;tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the&nbsp;tens column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ does actually represent $\\var{b2}$ since it is in the ones column)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the&nbsp;ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the ones&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tenths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\var{diff2}\\phantom{.}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1/10}$ since it is in the tenths column)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tenths&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tenths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;hundredths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;hundredths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;(note this&nbsp;$\\var{b0}$ actually represents&nbsp;$\\var{b0/100}$ since it is in the hundredths column)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths&nbsp;column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;hundredths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundredths&nbsp;column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we could keep adding zeros and continue the procedure but we only needed&nbsp;to determine&nbsp;the second decimal place in order to correctly round to one decimal place and so we now stop the procedure.</p>\n<p>Since the second decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span><span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$.&nbsp;Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (1 dec. pl.).</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, two digit divisor results in no remainder - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Nice easy case&nbsp;except&nbsp;divisor is a&nbsp;double&nbsp;digit. There is no remainder.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", 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"mod(floor(quotient1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#100))", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/100),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1,10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the thousands column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the hundreds column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the thousands column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the hundreds column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}\\,{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the hundreds column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tens column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the hundreds column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tens column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the ones column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We made it to exactly $\\var{b0}$, so there is no remainder in the ones column and no remainder at the end of the process!</p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^{\\var{diff1}}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p>Now we have run out of digits and we have no remainder. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p></p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the thousands column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{555}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the thousands column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{555}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the hundreds column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the&nbsp;hundreds column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundreds column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tens column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tens&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;ones column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><strong></strong></p>\n<p></p>\n<p>Now we have run out of digits and we have no remainder. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, two digit divisor results in a remainder which is expressed as a fraction - short or long division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is double&nbsp;digit. There is&nbsp;a remainder which we express as a fraction.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"newnum": {"name": "newnum", "group": "Ungrouped variables", "definition": "remainder/gcd", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "random(1..divisor1-1)", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/100),10)", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(11..49 except 20..40#10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1+remainder", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "gcd": {"name": "gcd", "group": "Ungrouped variables", "definition": "gcd(remainder,divisor1)", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/100),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1,10)", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/1000),10)", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "newden": {"name": "newden", "group": "Ungrouped variables", "definition": "divisor1/gcd", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#100))", "description": "<p>this is the quotient without reference to the remainder (so we don't get floating point errors)</p>", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "gcd", "newnum", "newden"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]] and&nbsp;[[1]]</p>\n<p></p>\n<p><strong>Note</strong>, your answer should involve a whole number and a fraction. Put the whole number in the first field and the fraction in the second. For example, if your answer was $4\\frac{3}{5}$ put $4$ in the first field and $3/5$ in the second field.</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the thousands column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the hundreds column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the thousands column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the hundreds column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}\\,{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the hundreds column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tens column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the hundreds column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tens column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the ones column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We only made it to $\\var{prod0}$ but we were aiming for $\\var{b0}$, so we were $\\var{diff0}$ away. This is the <em>remainder</em> (what remains to be divided).&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^{\\var{diff1}}{\\var{dd0}}}\\end{array} \\begin{array}{r} \\quad \\color{red}{\\text{r}\\,\\var{diff0}}\\\\\\,\\end{array}$</p>\n<p>Now we have run out of digits and we are left with&nbsp;a remainder of $\\var{remainder}$. Note this&nbsp;$\\var{remainder}$ hasn't been divided by $\\var{divisor1}$ yet (that's what remainder means!). Recall that dividing $\\var{remainder}$ by&nbsp;$\\var{divisor1}$ is the same as the fraction $\\frac{\\var{remainder}}{\\var{divisor1}}$<span data-jme-visible=\"gcd&lt;&gt;1\">&nbsp;which by removing the common factor of $\\var{gcd}$ from the top and the bottom is equivalent to the fraction $\\frac{\\var{newnum}}{\\var{newden}}$</span>.</p>\n<p></p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}+\\frac{\\var{newnum}}{\\var{newden}}$, which we&nbsp;usually write without the plus sign as the mixed number $\\var{quotient1}\\frac{\\var{newnum}}{\\var{newden}}$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p></p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The thousands column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*1000}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the thousands column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the thousands column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the thousands column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{555}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the thousands column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{555}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the hundreds column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundreds column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents&nbsp;$\\var{b2*100}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the hundreds column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the hundreds column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the&nbsp;hundreds column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundreds column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tens column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1*10}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tens column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tens&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the ones column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;ones column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{555}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{55}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we have run out of digits and we are left with&nbsp;a remainder of $\\var{remainder}$. Note this&nbsp;$\\var{remainder}$ hasn't been divided by $\\var{divisor1}$ yet (that's what remainder means!). Recall that dividing $\\var{remainder}$ by&nbsp;$\\var{divisor1}$ is the same as the fraction $\\frac{\\var{remainder}}{\\var{divisor1}}$<span data-jme-visible=\"gcd&lt;&gt;1\">&nbsp;which by removing the common factor of $\\var{gcd}$ from the top and the bottom is equivalent to the fraction $\\frac{\\var{newnum}}{\\var{newden}}$</span>.</p>\n<p></p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}+\\frac{\\var{newnum}}{\\var{newden}}$, which we&nbsp;usually write without the plus sign as the mixed number $\\var{quotient1}\\frac{\\var{newnum}}{\\var{newden}}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{remainder}/{divisor1}", "maxValue": "{remainder}/{divisor1}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, two digit divisor results in a remainder, answer expressed as a terminating decimal - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is a two digit number. There is a remainder which we express as a decimal by continuing the division process. No rounding is required by design (another question will include rounding off).</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(11..32 except [20,30])", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "0\n//random(1..divisor1-1)", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "divisor1*quotient1+remainder", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1*100,10)", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "random(ceil(3001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]&nbsp;</p>\n<p></p>\n<p><strong>Note</strong>, your answer should be a decimal.</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this $\\var{dd3}$ actually represents $\\var{dd3*10}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}}.\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}\\,.\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the ones column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,.\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,.\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\"</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,.\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tenths column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tenths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>Step 1:</strong></p>\n<p>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.1}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the hundredths column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the hundredths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundredths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.01}$)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong>We made it exactly to $\\var{b0}$ so there is no remainder.</p>\n<p></p>\n<p>Now we have run out of digits and we are left with no remainder.&nbsp;</p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>&nbsp;&nbsp;</p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;tens column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*10}$ since it is in the tens column)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the&nbsp;tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the&nbsp;tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the&nbsp;tens column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ does actually represent $\\var{b2}$ since it is in the ones column)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the&nbsp;ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the ones&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tenths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\var{diff2}\\phantom{.}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1/10}$ since it is in the tenths column)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tenths&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tenths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;hundredths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;hundredths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;(note this&nbsp;$\\var{b0}$ actually represents&nbsp;$\\var{b0/100}$ since it is in the hundredths column)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths&nbsp;column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;hundredths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundredths&nbsp;column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we have run out of digits and we are left with no remainder.&nbsp;</p>\n<p>Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{quotient1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "quotient1", "maxValue": "quotient1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division, two digit divisor results in a remainder, answer expressed as a decimal rounded to 1 decimal place - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Divisor is a two-digit number. There is a remainder which we express as a decimal by continuing the division process. Rounding is required to one decimal place. The working suggests determining the second decimal place so the student knows whether to round up or down.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(quotient1,1)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(11..32 except [20,30])", "description": "<p>excluded 5 so that the decimal part is longer than 1 place.</p>", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}, "qd0": 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"can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1*100,10)", "description": "", "templateType": "anything", "can_override": false}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "<p>qd2</p>", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "dividend1/divisor1\n//random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "mod(dividend1,divisor1)", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dividend1}\\div\\var{divisor1}=$[[0]]&nbsp;(1 decimal place)</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>Actually, since we want the answer to <strong>one decimal place</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have <strong>two decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}}$</p>\n<p>Why two? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The tens column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this $\\var{dd3}$ actually represents $\\var{dd3*10}$)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{^\\var{diff3}\\var{qd2}}.\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}^\\phantom{\\var{diff3}}\\var{dd2}\\,.\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the tens column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the ones column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the tens column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the ones column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,.\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,.\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\"</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\,.\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the tenths column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the tenths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>Step 1: </strong>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.1}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the hundredths column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the hundredths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundredths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.01}$)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,.\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,^{\\var{diff3}}{\\var{dd2}}\\,.\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p><strong>Step 2:&nbsp;</strong>We could work out the remainder, we could keep adding zeros and continue the procedure but we only needed&nbsp;to determine&nbsp;the second decimal place in order to correctly round to one decimal place and so we now stop the procedure.</p>\n<p>Since the second decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span><span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$.&nbsp;Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (1 dec. pl.).</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!&nbsp;</p>\n<p>Actually, since we want the answer to <strong>one decimal place</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have <strong>two decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}}$</p>\n<p>Why two? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p></p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;tens column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this&nbsp;$\\var{dd3}$ actually represents&nbsp;$\\var{dd3*10}$ since it is in the tens column)&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the&nbsp;tens column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the&nbsp;tens column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the tens column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the&nbsp;tens column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tens column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{5.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the&nbsp;ones column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;ones column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ does actually represent $\\var{b2}$ since it is in the ones column)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the&nbsp;ones column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the ones&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{.55}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the&nbsp;tenths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\var{diff2}\\phantom{.}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1/10}$ since it is in the tenths column)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the tenths&nbsp;column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;tenths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;hundredths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;hundredths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;(note this&nbsp;$\\var{b0}$ actually represents&nbsp;$\\var{b0/100}$ since it is in the hundredths column)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths&nbsp;column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;hundredths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the&nbsp;hundredths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the&nbsp;hundredths&nbsp;column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we could keep adding zeros and continue the procedure but we only needed&nbsp;to determine&nbsp;the second decimal place in order to correctly round to one decimal place and so we now stop the procedure.</p>\n<p>Since the second decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span><span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$.&nbsp;Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (1 dec. pl.).</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Working with decimals", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Decimals: how to read", "Decimals: common misconceptions", "Decimals: Addition", "Decimals: Subtraction", "Decimals: Multiplying and dividing by powers of ten", "Decimals: Multiplication", "Decimals: division (includes rounding the answer) - long or short division", "Decimals: rounding"], "variable_overrides": [[], [], [], [], [], [], [], []], "questions": [{"name": "Merryn's copy of Decimals: how to read", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. 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\"fifteen\"],\n[dpformat(0.16, 2), \"one six\", \"sixteen\"],\n[dpformat(0.17, 2), \"one seven\", \"seventeen\"],\n[dpformat(0.18, 2), \"one eight\", \"eighteen\"],\n[dpformat(0.19, 2), \"one nine\", \"ninteen\"], \n\n[dpformat(0.20, 2), \"two zero\", \"twenty\"],  \n[dpformat(0.21, 2), \"two one\", \"twenty one\"],  \n[dpformat(0.22, 2), \"two two\", \"twenty two\"],\n[dpformat(0.23, 2), \"two three\", \"twenty three\"],\n[dpformat(0.24, 2), \"two four\", \"twenty four\"], \n[dpformat(0.25, 2), \"two five\", \"twenty five\"],\n[dpformat(0.26, 2), \"two six\", \"twenty six\"],\n[dpformat(0.27, 2), \"two seven\", \"twenty seven\"],\n[dpformat(0.28, 2), \"two eight\", \"twenty eight\"],\n[dpformat(0.29, 2), \"two nine\", \"twenty nine\"],   \n  \n[dpformat(0.30, 2), \"three zero\", \"thirty\"],  \n[dpformat(0.31, 2), \"three one\", \"thirty one\"],  \n[dpformat(0.32, 2), \"three two\", \"thirty two\"],\n[dpformat(0.33, 2), \"three three\", \"thirty three\"],\n[dpformat(0.34, 2), \"three four\", \"thirty four\"], \n[dpformat(0.35, 2), \"three five\", \"thirty five\"],\n[dpformat(0.36, 2), \"three six\", \"thirty six\"],\n[dpformat(0.37, 2), \"three seven\", \"thirty seven\"],\n[dpformat(0.38, 2), \"three eight\", \"thirty eight\"],\n[dpformat(0.39, 2), \"three nine\", \"thirty nine\"],  \n  \n[dpformat(0.40, 2), \"four zero\", \"forty\"],  \n[dpformat(0.41, 2), \"four one\", \"forty one\"],  \n[dpformat(0.42, 2), \"four two\", \"forty two\"],\n[dpformat(0.43, 2), \"four three\", \"forty three\"],\n[dpformat(0.44, 2), \"four four\", \"forty four\"], \n[dpformat(0.45, 2), \"four five\", \"forty five\"],\n[dpformat(0.46, 2), \"four six\", \"forty six\"],\n[dpformat(0.47, 2), \"four seven\", \"forty seven\"],\n[dpformat(0.48, 2), \"four eight\", \"forty eight\"],\n[dpformat(0.49, 2), \"four nine\", \"forty nine\"], \n  \n[dpformat(0.50, 2), \"five zero\", \"fifty\"],  \n[dpformat(0.51, 2), \"five one\", \"fifty one\"],  \n[dpformat(0.52, 2), \"five two\", \"fifty two\"],\n[dpformat(0.53, 2), \"five three\", \"fifty three\"],\n[dpformat(0.54, 2), \"five four\", \"fifty four\"], \n[dpformat(0.55, 2), \"five five\", \"fifty five\"],\n[dpformat(0.56, 2), \"five six\", \"fifty six\"],\n[dpformat(0.57, 2), \"five seven\", \"fifty seven\"],\n[dpformat(0.58, 2), \"five eight\", \"fifty eight\"],\n[dpformat(0.59, 2), \"five nine\", \"fifty nine\"], \n  \n[dpformat(0.60, 2), \"six zero\", \"sixty\"],  \n[dpformat(0.70, 2), \"seven zero\", \"seventy\"],  \n[dpformat(0.80, 2), \"eight zero\", \"eighty\"],  \n[dpformat(0.90, 2), \"nine zero\", \"ninety\"],  \n)]+[random(\n[dpformat(0.100, 3), \"one zero zero\", \"one hundred\"],  \n[dpformat(0.200, 3), \"two zero zero\", \"two hundred\"],  \n[dpformat(0.300, 3), \"three zero zero\", \"three hundred\"],\n[dpformat(0.400, 3), \"four zero zero\", \"four hundred\"],\n[dpformat(0.500, 3), \"five zero zero\", \"five hundred\"], \n[dpformat(0.600, 3), \"six zero zero\", \"six hundred\"],\n[dpformat(0.700, 3), \"seven zero zero\", \"seven hundred\"],\n[dpformat(0.800, 3), \"eight zero zero\", \"eight hundred\"],\n[dpformat(0.900, 3), \"nine zero zero\", \"nine hundred\"],\n[dpformat(0.120, 3), \"one two zero\", \"one hundred and twenty\"],  \n[dpformat(0.230, 3), \"two three zero\", \"two hundred and thirty\"],  \n[dpformat(0.340, 3), \"three four zero\", \"three hundred and forty\"],\n[dpformat(0.450, 3), \"four five zero\", \"four hundred and fifty\"],\n[dpformat(0.501, 3), \"five zero one\", \"five hundred and one\"], \n[dpformat(0.602, 3), \"six zero two\", \"six hundred and two\"],\n[dpformat(0.703, 3), \"seven zero three\", \"seven hundred and three\"],\n[dpformat(0.804, 3), \"eight zero four\", \"eight hundred and four\"],\n[dpformat(0.905, 3), \"nine zero five\", \"nine hundred and five\"]\n  )])\n ", "description": "", "templateType": "anything", "can_override": false}, "placedig": {"name": "placedig", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "hundredths": {"name": "hundredths", "group": "Ungrouped variables", "definition": "if(len(pron[0][0])=4,dec(pron[0][0])*100,dec(pron[1][0])*100)", "description": "", "templateType": "anything", "can_override": false}, "pron2": {"name": "pron2", "group": "Ungrouped variables", "definition": "random(\n[dpformat(0.100, 3), \"zero point one zero zero\", \"zero point one hundred\"],  \n[dpformat(0.200, 3), \"zero point two zero zero\", \"zero point two hundred\"],  \n[dpformat(0.300, 3), \"zero point three zero zero\", \"zero point three hundred\"],\n[dpformat(0.400, 3), \"zero point four zero zero\", \"zero point four hundred\"],\n[dpformat(0.500, 3), \"zero point five zero zero\", \"zero point five hundred\"], \n[dpformat(0.600, 3), \"zero point six zero zero\", \"zero point six hundred\"],\n[dpformat(0.700, 3), \"zero point seven zero zero\", \"zero point seven hundred\"],\n[dpformat(0.800, 3), \"zero point eight zero zero\", \"zero point eight hundred\"],\n[dpformat(0.900, 3), \"zero point nine zero zero\", \"zero point nine hundred\"],\n[dpformat(0.120, 3), \"zero point one two zero\", \"zero point one hundred and twenty\"],  \n[dpformat(0.230, 3), \"zero point two three zero\", \"zero point two hundred and thirty\"],  \n[dpformat(0.340, 3), \"zero point three four zero\", \"zero point three hundred and forty\"],\n[dpformat(0.450, 3), \"zero point four five zero\", \"zero point four hundred and fifty\"],\n[dpformat(0.501, 3), \"zero point five zero one\", \"zero point five hundred and one\"], \n[dpformat(0.602, 3), \"zero point six zero two\", \"zero point six hundred and two\"],\n[dpformat(0.703, 3), \"zero point seven zero three\", \"zero point seven hundred and three\"],\n[dpformat(0.804, 3), \"zero point eight zero four\", \"zero point eight hundred and four\"],\n[dpformat(0.905, 3), \"zero point nine zero five\", \"zero point nine hundred and five\"]\n  )", "description": "", "templateType": "anything", "can_override": false}, "thousandths": {"name": "thousandths", "group": "Ungrouped variables", "definition": "if(len(pron[0][0])=5,dec(pron[0][0])*1000,dec(pron[1][0])*1000)", "description": "", "templateType": "anything", "can_override": false}, "identify": {"name": "identify", "group": "Ungrouped variables", "definition": "if(len(pron[0][0])=5,[[1000,dec(pron[0][0])*1000],[100, dec(pron[1][0])*100]],[[100, dec(pron[0][0])*100],[1000,dec(pron[1][0])*1000]])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["placedig", "place", "pron", "identify", "thousandths", "hundredths", "pron2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The digit $\\var{placedig}$ in the decimal $\\var{place[0]}$ represents&nbsp;&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: left;\">The decimal 0.1 is also known as \"one tenth\" (notice you need ten of them to make a whole).</p>\n<p>The decimal 0.01 is also known as \"one hundredth\" (notice you need a hundred&nbsp;of them to make a whole).</p>\n<p>The decimal 0.001 is also known as \"one thousandth\" (notice you need a thousand&nbsp;of them to make a whole).</p>\n<p></p>\n<hr/>\n<p></p>\n<p>That is,&nbsp;the digit $\\var{placedig}$ in the decimal $\\var{place[0]}$ is in the {place[3]} column and so represents $\\var{placedig}$ {place[3]}.</p>"}], "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$\\var{placedig}$ {place[1]}</p>", "<p>$\\var{placedig}$ {place[2]}</p>", "<p>$\\var{placedig}$ {place[3]}</p>"], "matrix": [0, 0, "1"], "distractors": ["", "", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The decimal $\\var{pron[0][0]}$ should be read as [[0]]</p>\n<p>or, as $\\var{identify[0][1]}$ [[1]].</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Say each&nbsp;digit individually&nbsp;after the decimal point.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>It makes no sense to call 0.500, \"zero point five hundred\"&nbsp;since that sounds a lot bigger than \"zero <g class=\"gr_ gr_9 gr-alert gr_gramm Grammar multiReplace\" id=\"9\" data-gr-id=\"9\">point</g> five\", or \"zero <g class=\"gr_ gr_10 gr-alert gr_gramm Grammar multiReplace\" id=\"10\" data-gr-id=\"10\">point</g> fifty\", but these are all equal to the same number! Pronouncing decimals like this is misleading and doesn't help with your intuition. However, this decimal is 500 of something, it is 500 <em>thousandths!</em> But be careful, even reading it that way can be ambiguous when it's read aloud.&nbsp;</p>\n<p></p>\n<hr/>\n<p></p>\n<p>That is, $\\var{pron[0][0]}$ is read as <em>{pron[0][1]}</em>.</p>\n<p>Alternatively, since the last digit written is in the <span data-jme-visible=\"identify[0][0]=100\"><em>hundredths</em></span><span data-jme-visible=\"identify[0][0]=1000\"><em>thousandths</em></span> column, we can think of this (and read it out) as&nbsp;$\\var{identify[0][1]}$ <span data-jme-visible=\"identify[0][0]=100\">hundredths.</span><span data-jme-visible=\"identify[0][0]=1000\">thousandths.</span>&nbsp;</p>"}], "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>zero point {pron[0][1]}</p>", "<p>zero point {pron[0][2]}</p>"], "matrix": ["1", 0], "distractors": ["", "Please see the steps"]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["tenths", "hundredths", "thousandths"], "matrix": [0, "if(identify[0][0]=100,1,0)", "if(identify[0][0]=1000,1,0)"], "distractors": ["", "", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The decimal $\\var{pron[1][0]}$ should be read as [[0]]</p>\n<p>or, as $\\var{identify[1][1]}$ [[1]].</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Say each&nbsp;digit individually&nbsp;after the decimal point.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>It makes no sense to call 0.500, \"zero point five hundred\"&nbsp;since that sounds a lot bigger than \"zero <g class=\"gr_ gr_9 gr-alert gr_gramm Grammar multiReplace\" id=\"9\" data-gr-id=\"9\">point</g> five\", or \"zero <g class=\"gr_ gr_10 gr-alert gr_gramm Grammar multiReplace\" id=\"10\" data-gr-id=\"10\">point</g> fifty\", but these are all equal to the same number! Pronouncing decimals like this is misleading and doesn't help with your intuition. However, this decimal is 500 of something, it is 500 <em>thousandths!</em> But be careful, even reading it that way can be ambiguous when it's read aloud.&nbsp;</p>\n<p></p>\n<hr/>\n<p></p>\n<p>That is, $\\var{pron[1][0]}$ is read as <em>{pron[1][1]}</em>.</p>\n<p>Alternatively, since the last digit written is in the <span data-jme-visible=\"identify[1][0]=100\"><em>hundredths</em></span><span data-jme-visible=\"identify[1][0]=1000\"><em>thousandths</em></span> column, we can think of this as $\\var{identify[1][1]}$ <span data-jme-visible=\"identify[1][0]=100\">hundredths.</span><span data-jme-visible=\"identify[1][0]=1000\">thousandths.</span>&nbsp;</p>"}], "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>zero point {pron[1][1]}</p>", "<p>zero point {pron[1][2]}</p>"], "matrix": ["1", 0], "distractors": ["", "Please see the steps"]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["tenths", "hundredths", "thousandths"], "matrix": [0, "if(identify[1][0]=100,1,0)", "if(identify[1][0]=1000,1,0)"], "distractors": ["", "", ""]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: common misconceptions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Some students believe a decimal is larger if it is longer, some believe a decimal is larger if its first non-zero digit is larger.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Use the drop-down menu to&nbsp;create&nbsp;the correct sentence.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"fnzdigbig": {"name": "fnzdigbig", "group": "Ungrouped variables", "definition": "random(fnzdigsmall+2..9)", "description": "", "templateType": "anything", "can_override": false}, "trailshort": {"name": "trailshort", "group": "Ungrouped variables", "definition": "random(0.1..0.9#0.1)", "description": "", "templateType": "anything", "can_override": false}, "fnz": {"name": "fnz", "group": "Ungrouped variables", "definition": "random([[fnzdigsmall/10,fnzdigbig/100,1,0],[fnzdigbig/100,fnzdigsmall/10,0,1]])", "description": "", "templateType": "anything", "can_override": false}, "trail": {"name": "trail", "group": "Ungrouped variables", "definition": "shuffle(['\\$\\\\var{trailshort}\\$','\\$\\\\var{trailshort}0\\$','\\$\\\\var{trailshort}00\\$','\\$\\\\var{trailshort}000\\$'])[0..2]", "description": "", "templateType": "anything", "can_override": false}, "fnzdigsmall": {"name": "fnzdigsmall", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "<p>first non-zero digit</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["trail", "trailshort", "fnzdigsmall", "fnzdigbig", "fnz"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The number {trail[0]} is&nbsp;[[0]]&nbsp;{trail[1]}</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The trailing zeros do not change the value of a decimal. In the same way that $42$ is no different to $000042$ (regardless of how many zeros are placed at the front), $\\var{trailshort}$ is no different to $\\var{trailshort}0000$ (regardless of how many zeros are placed at the back). This is why it is important to read things such as $0.200$ as \"zero <g class=\"gr_ gr_9 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"9\" data-gr-id=\"9\">point</g> two <g class=\"gr_ gr_10 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling only-del replaceWithoutSep\" id=\"10\" data-gr-id=\"10\">zero zero</g>\" and not as \"zero point two hundred\".</p>\n<p>In general, the<em> length</em> or number of digits in <g class=\"gr_ gr_14 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar only-del replaceWithoutSep\" id=\"14\" data-gr-id=\"14\">a decimal</g> does not tell us anything about how <em>big</em> the decimal is. The only things that affect&nbsp;the actual value of a decimal are the non-zero digits and their placement relative to the decimal point (that is their face value and place value).</p>"}], "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["greater than", "less than", "equal to"], "matrix": [0, 0, "1"], "distractors": ["", "", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The number $\\var{fnz[0]}$ is&nbsp;[[0]]&nbsp;$\\var{fnz[1]}$</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><span data-jme-visible=\"fnz[2]=0\">You may have suspected that $\\var{fnz[0]}$ was&nbsp;greater than $\\var{fnz[1]}$ simply because $\\var{fnzdigbig}$ was greater than $\\var{fnzdigsmall}$, however, $\\var{fnzdigbig}$ is in a column with a smaller place value!</span></p>\n<p><span data-jme-visible=\"fnz[2]=1\">You may have suspected that $\\var{fnz[0]}$ was less than $\\var{fnz[1]}$ simply because $\\var{fnzdigsmall}$ was&nbsp;less than $\\var{fnzdigbig}$, however, $\\var{fnzdigsmall}$ is in a column with a larger place value!</span></p>\n<p>In general, the<em>&nbsp;first non-zero digit</em>&nbsp;does not tell us anything about how <em>big</em> the decimal is. The only things that affect&nbsp;the actual value of a decimal are the non-zero digits and their placement relative to the decimal point (that is their face value and place value).</p>\n<p>You can add zeros so that the decimals have the same number of decimals places, and then, comparing them might be easier. That is, by&nbsp;appending a zero (which doesn't affect the value) onto the end of $\\var{fnzdigsmall/10}$ it might be clearer that $\\var{fnzdigsmall/10}0$ is greater than $\\var{fnzdigbig/100}$. Note that $\\var{fnzdigsmall/10}0$ is $\\var{fnzdigsmall}0$ hundredths whereas $\\var{fnzdigbig/100}$ is $\\var{fnzdigbig}$ hundredths.</p>"}], "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["greater than", "less than", "equal to"], "matrix": ["fnz[2]", "fnz[3]", "0"], "distractors": ["", "", ""]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: Addition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Decimals addition algorithm. 2 and 3 digit numbers. Carrying.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you click on <em>Show steps</em> it will show you the full working using the standard method. Click on <em>Try another question like this one</em> to get a new set of numbers to add.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"chunsumlastdigit": {"name": "chunsumlastdigit", "group": "Ungrouped variables", "definition": "mod(chunsum,10)", "description": "", "templateType": "anything", "can_override": false}, "threedigit1": {"name": "threedigit1", "group": "Ungrouped variables", "definition": "cdigs[0]/1000+cdigs[1]/100+cdigs[2]/10", "description": "", "templateType": "anything", "can_override": false}, "cunitsum": {"name": "cunitsum", "group": "Ungrouped variables", "definition": "cdigs[0]+cdigs[3]", "description": "", "templateType": "anything", "can_override": false}, "ctencarry": {"name": "ctencarry", "group": "Ungrouped variables", "definition": "floor(ctensum/10)", "description": "", "templateType": "anything", "can_override": false}, "cunitsumlastdigit": {"name": "cunitsumlastdigit", "group": "Ungrouped variables", "definition": "mod(cunitsum,10)", "description": "", "templateType": "anything", "can_override": false}, "cunitcarry": {"name": "cunitcarry", "group": "Ungrouped variables", "definition": "floor(cunitsum/10)", "description": "", "templateType": "anything", "can_override": false}, "chunsum": {"name": "chunsum", "group": "Ungrouped variables", "definition": "ctencarry+cdigs[2]+cdigs[5]", "description": "", "templateType": "anything", "can_override": false}, "ctensum": {"name": "ctensum", "group": "Ungrouped variables", "definition": "cdigs[1]+cdigs[4]+cunitcarry", "description": "", "templateType": "anything", "can_override": false}, "threedigit2": {"name": "threedigit2", "group": "Ungrouped variables", "definition": "cdigs[3]/1000+cdigs[4]/100+cdigs[5]/10", "description": "", "templateType": "anything", "can_override": false}, "ctensumlastdigit": {"name": "ctensumlastdigit", "group": "Ungrouped variables", "definition": "mod(ctensum,10)", "description": "", "templateType": "anything", "can_override": false}, "chuncarry": {"name": "chuncarry", "group": "Ungrouped variables", "definition": "floor(chunsum/10)", "description": "", "templateType": "anything", "can_override": false}, "cans": {"name": "cans", "group": "Ungrouped variables", "definition": "threedigit1+threedigit2", "description": "", "templateType": "anything", "can_override": false}, "cdigs": {"name": "cdigs", "group": "Ungrouped variables", "definition": "[0]+shuffle(3..9)", "description": "", "templateType": "anything", "can_override": false}, "cunitsumtensdigit": {"name": "cunitsumtensdigit", "group": "Ungrouped variables", "definition": "0.1*(cunitsum-cunitsumlastdigit)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["cdigs", "threedigit1", "threedigit2", "cans", "cunitsum", "cunitsumlastdigit", "cunitcarry", "ctensum", "ctensumlastdigit", "ctencarry", "chunsum", "chunsumlastdigit", "chuncarry", "cunitsumtensdigit"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{threedigit1}+\\var{threedigit2} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally, we set up $\\var{threedigit1}+\\var{threedigit2}$ with the decimal points lined up vertically so that the columns with the same place value are also lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[2]}$</td>\n<td>$\\var{cdigs[1]}$</td>\n<td></td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Note that we can <em>pad out</em> the decimal with zeros if we prefer:</p>\n<table>\n<tbody>\n<tr>\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[2]}$</td>\n<td>$\\var{cdigs[1]}$</td>\n<td>$\\color{red}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we add the digits in the column to the far right (in this case, the <em>thousandths</em> column).</p>\n<p><span data-jme-visible=\"cunitsumtensdigit=0\">This results in&nbsp;$\\var{cunitsum}$ and so&nbsp;we place $\\var{cunitsumlastdigit}$ under the line in this column. </span></p>\n<p><span data-jme-visible=\"cunitsumtensdigit=1\">This results in&nbsp;$\\var{cunitsum}$ and so&nbsp;we place $\\var{cunitsumlastdigit}$ under the line in this column and carry the $1$ into the&nbsp;next column to the left.&nbsp;</span></p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{0}$</td>\n<td>.</td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{\\color{red}1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\color{green}{\\overset{\\phantom{1}}{\\var{cdigs[0]}}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\color{green}{\\var{cdigs[3]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the&nbsp;next&nbsp;column to the left (in this case, the <em>hundredths</em> column).</p>\n<p><span data-jme-visible=\"ctencarry=0\">This results in&nbsp;$\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in&nbsp;this column.</span></p>\n<p><span data-jme-visible=\"ctencarry=1\">This results in&nbsp;$\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in this column and carry the $1$ into the next column to the left.</span></p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{0}$</td>\n<td>.</td>\n<td><span data-jme-visible=\"ctencarry=1\">$\\overset{\\color{red}{1}}{\\var{cdigs[2]}}$</span> <span data-jme-visible=\"ctencarry=0\">$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</span></td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\color{green}{\\overset{1}{\\var{cdigs[1]}}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\color{green}{\\overset{\\phantom{0}}{\\var{cdigs[1]}}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\color{green}{\\var{cdigs[4]}}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ctenSumlastdigit}}$</td>\n<td>${\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>Now we add the digits in the&nbsp;next&nbsp;column to the left (in this case, the&nbsp;<em>tenths</em>&nbsp;column).</p>\n<p><span data-jme-visible=\"cans&lt;1\">This is $\\var{chunsum}$ so we place&nbsp;$\\var{chunsum}$ under the line in&nbsp;this column.</span></p>\n<p><span data-jme-visible=\"cans&gt;0.999\">This is $\\var{chunsum}$ so we place&nbsp;$\\var{chunsumlastdigit}$ under the line in this column and carry $\\var{chuncarry}$ into the next column to the left (which in this case is the ones column on the other side of the decimal point).</span></p>\n<table>\n<tbody>\n<tr>\n<td><span data-jme-visible=\"chuncarry&gt;0\">$\\overset{\\color{red}1}{0}$</span> <span data-jme-visible=\"chuncarry=0\">$\\overset{\\phantom{1}}{0}$</span></td>\n<td>.</td>\n<td><span data-jme-visible=\"ctencarry=1\">$\\color{green}{\\overset{1}{\\var{cdigs[2]}}}$</span> <span data-jme-visible=\"ctencarry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{cdigs[2]}}}$</span></td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\color{green}{\\var{cdigs[5]}}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>.</td>\n<td>$\\color{red}{\\var{chunsumlastdigit}}$</td>\n<td>$\\var{ctenSumlastdigit}$</td>\n<td>${\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we add the digits in the&nbsp;next&nbsp;column to the left (in this case, the&nbsp;<em>ones</em>&nbsp;column).</p>\n<p><span data-jme-visible=\"cans&lt;1\">This is&nbsp;just $0$ so we place&nbsp;$0$ under the line in&nbsp;this column.</span></p>\n<p><span data-jme-visible=\"cans&gt;0.999\">This is just $\\var{chuncarry}$ so we place&nbsp;$\\var{chuncarry}$ under the line in this column.</span></p>\n<table>\n<tbody>\n<tr>\n<td><span data-jme-visible=\"chuncarry&gt;0\">$\\color{green}{\\overset{1}{0}}$</span> <span data-jme-visible=\"chuncarry=0\">$\\color{green}{\\overset{\\phantom{1}}{0}}$</span></td>\n<td>.</td>\n<td><span data-jme-visible=\"ctencarry=1\">$\\overset{1}{\\var{cdigs[2]}}$</span> <span data-jme-visible=\"ctencarry=0\">$\\overset{\\phantom{1}}{\\var{cdigs[2]}}$</span></td>\n<td><span data-jme-visible=\"cunitcarry=1\">$\\overset{1}{\\var{cdigs[1]}}$</span> <span data-jme-visible=\"cunitcarry=0\">$\\overset{\\phantom{0}}{\\var{cdigs[1]}}$</span></td>\n<td>$\\overset{\\phantom{1}}{\\var{cdigs[0]}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$\\color{green}{0}$</td>\n<td>.</td>\n<td>$\\var{cdigs[5]}$</td>\n<td>$\\var{cdigs[4]}$</td>\n<td>$\\var{cdigs[3]}$</td>\n<td></td>\n</tr>\n<tr>\n<td>$\\color{red}{\\var{chuncarry}}$</td>\n<td>.</td>\n<td>$\\var{chunsumlastdigit}$</td>\n<td>$\\var{ctenSumlastdigit}$</td>\n<td>${\\var{cunitSumLastDigit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p>The answer is therefore $\\var{cans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "cans", "maxValue": "cans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: Subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Subtracting a decimal with 3 decimal places from a decimal with 2 or 3 decimal places. borrowing is necessary. This was modified from a subtraction question using integers with each number divided by 1000 so the variables have names referring to ones, tens, hundreds etc.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on <em>Show steps</em> to see the full working. Then, once you understand how to do the question, click on <em>Try another question like this one</em> to start again.</p>", "advice": "<p class=\"p1\">Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p class=\"p1\">If you are unsure of how to do&nbsp;a question, click on&nbsp;<i>Show steps</i>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<i>Try another question like this one</i>&nbsp;to start again.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"tendiff": {"name": "tendiff", "group": "c", "definition": "if(unitdiff>=0,top[1]-bot[1],top[1]-1-bot[1])", "description": "", "templateType": "anything", "can_override": false}, "hundiff": {"name": "hundiff", "group": "c", "definition": "if(tendiff>=0,top[2]-bot[2],top[2]-1-bot[2])", "description": "", "templateType": "anything", "can_override": false}, "newtopten": {"name": "newtopten", "group": "c", "definition": "if(unitdiff>=0,top[1],top[1]-1)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "c", "definition": "topnum-botnum", "description": "", "templateType": "anything", "can_override": false}, "newtophun": {"name": "newtophun", "group": "c", "definition": "if(tendiff>=0,top[2],top[2]-1)", "description": "", "templateType": "anything", "can_override": false}, "anshun": {"name": "anshun", "group": "c", "definition": "mod(floor(ans*10),10)", "description": "", "templateType": "anything", "can_override": false}, "topnum": {"name": "topnum", "group": "c", "definition": "top[0]/1000+top[1]/100+top[2]/10", "description": "", "templateType": "anything", "can_override": false}, "ansunit": {"name": "ansunit", "group": "c", "definition": "mod(ans*1000,10)", "description": "", "templateType": "anything", "can_override": false}, "unitdiff": {"name": "unitdiff", "group": "c", "definition": "top[0]-bot[0]", "description": "", "templateType": "anything", "can_override": false}, "bot": {"name": "bot", "group": "c", "definition": "if(top[0]<>0, \n  random(\n    [0, random(top[1]+1..9), random(1..top[2]-1)],\n    [random(top[0]+1..9), random(0..9), random(1..top[2]-1)]), \n [random(1..9), random(2..9), 1])\n\n//original \n//random(\n//if(top[0]<9,[random(top[0]+1..9), random(top[1]..9), random(1..top[2]-1)],if(top[1]<9,[random(top[0]..9), random(top[1]+1..9), random(1..top[2]-1)]),\"error\"),\n//if(top[1]<9,[random(0..top[0]), random(top[1]+1..9), random(1..top[2]-1)],if(top[1]=9,[random(top[0]..9), random(0..9), random(1..top[2]-1)]),\"error\")\n//)\n\n", "description": "<p>This should force some borrowing and <g class=\"gr_ gr_4 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"4\" data-gr-id=\"4\">paying</g> back, and that the&nbsp;final answer&nbsp;is positive.</p>", "templateType": "anything", "can_override": false}, "top": {"name": "top", "group": "c", "definition": "random([random(1..8),random(1..8),random(2..9)],[0,random(1..9),random(2..9)])", "description": "<p>the digits of a 2 or 3 decimal place number</p>", "templateType": "anything", "can_override": false}, "botnum": {"name": "botnum", "group": "c", "definition": "bot[0]/1000+bot[1]/100+bot[2]/10", "description": "", "templateType": "anything", "can_override": false}, "ansten": {"name": "ansten", "group": "c", "definition": "mod(floor(ans*100),10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": [], "variable_groups": [{"name": "c", "variables": ["top", "bot", "topnum", "botnum", "ans", "unitdiff", "tendiff", "hundiff", "ansunit", "ansten", "anshun", "newtopten", "newtophun"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{topnum}-\\var{botnum} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Generally&nbsp;we set up $\\var{topnum}-\\var{botnum}$ with the decimal points lined up vertically so that the columns with the same place value are also lined up vertically. <span data-jme-visible=\"top[0]=0 or bot[0]=0\">We also&nbsp;<em>pad out</em> the decimals with zeros:</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{top[2]}$</td>\n<td>$\\var{top[1]}$</td>\n<td><span data-jme-visible=\"top[0]=0\">$\\color{red}{\\var{top[0]}}$</span> <span data-jme-visible=\"top[0]&lt;&gt;0\">$\\var{top[0]}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td><span data-jme-visible=\"bot[0]=0\">$\\color{red}{\\var{bot[0]}}$</span> <span data-jme-visible=\"bot[0]&lt;&gt;0\">$\\var{bot[0]}$</span></td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>.</td>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now&nbsp;we try to subtract the digits in the column to the far right (in this case, the&nbsp;<em>thousandths</em>&nbsp;column).</p>\n<p><span data-jme-visible=\"unitdiff&gt;=0\">Since this is $\\var{ansunit}$ we&nbsp;write $\\var{ansunit}$ under the line in&nbsp;this column.</span></p>\n<p><span data-jme-visible=\"unitdiff&lt;0\">Since we can't&nbsp;take $\\var{bot[0]}$ away from $\\var{top[0]}$ (without using negative numbers) we <em>borrow</em>&nbsp;from the next column to the left (in this case, the <em>hundredths</em> column). This means we cross out the $\\var{top[1]}$ in the&nbsp;hundredths column&nbsp;and replace it with a $\\var{top[1]-1}$, and the $\\var{top[0]}$ becomes a $\\var{10+top[0]}$. Now we can do&nbsp;$\\var{10+top[0]}-\\var{bot[0]}$, and write the result, $\\var{ansunit}$, under the line in the thousandths column.</span>&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{0}$</td>\n<td>.</td>\n<td>$\\overset{\\phantom{1}}{\\var{top[2]}}$</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\overset{\\color{red}{\\var{newtopten}}}{\\var{top[1]}\\mkern-7.5mu\\color{red}/}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\color{green}{\\var{top[0]}}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\color{green}{\\var{top[0]}}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\color{green}{\\var{bot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>.</td>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ansunit}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the hundredths column.</p>\n<p><span data-jme-visible=\"tendiff&gt;=0\">Since this is $\\var{ansten}$ we&nbsp;write $\\var{ansten}$ under the line in&nbsp;this column.</span></p>\n<p><span data-jme-visible=\"tendiff&lt;0\">Since we can't&nbsp;take $\\var{bot[1]}$ away from $\\var{newtopten}$ (without using negative numbers) we <em>borrow</em>&nbsp;from the next column to the left (in this case, the <em>tenths</em> column). This means we cross out the $\\var{top[2]}$ in the&nbsp;tenths column&nbsp;and replace it with a $\\var{top[2]-1}$, and the $\\var{newtopten}$ in the&nbsp;hundredths column becomes a $\\var{10+newtopten}$. Now we can do&nbsp;$\\var{10+newtopten}-\\var{bot[1]}$, and write the result, $\\var{ansten}$, under the line in the hundredths.</span>&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{0}$</td>\n<td>.</td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\color{red}{\\var{newtophun}}}{\\var{top[2]}\\mkern-7.5mu\\color{red}{/}}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[2]}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{\\color{red}{1}\\color{green}{\\var{newtopten}}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\color{green}{\\var{newtopten}}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">$\\color{red}{^1}\\overset{\\phantom{1}}{\\color{green}{\\var{top[1]}}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\color{green}{\\var{top[1]}}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$0$</td>\n<td>.</td>\n<td>$\\var{bot[2]}$</td>\n<td>$\\color{green}{\\var{bot[1]}}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>.</td>\n<td></td>\n<td>$\\color{red}{\\var{ansten}}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Now we&nbsp;try to subtract the digits in the&nbsp;tenths column and&nbsp;then subtract the digits in the ones column.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>$\\overset{\\phantom{1}}{\\color{green}{0}}$</td>\n<td>.</td>\n<td>\n<p><span data-jme-visible=\"tendiff&lt;0\">$\\overset{\\color{green}{\\var{newtophun}}}{\\var{top[2]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\color{green}{\\var{top[2]}}}$</span></p>\n</td>\n<td>\n<p><span data-jme-visible=\"unitdiff&lt;0 and tendiff&lt;0\">$\\overset{{1}\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&lt;0 and tendiff&gt;=0\">$\\overset{\\var{newtopten}}{\\var{top[1]}\\mkern-7.5mu/}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[1]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0 and tendiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[1]}}$</span></p>\n</td>\n<td><span data-jme-visible=\"unitdiff&lt;0\">${^1}\\overset{\\phantom{1}}{\\var{top[0]}}$</span> <span data-jme-visible=\"unitdiff&gt;=0\">$\\overset{\\phantom{1}}{\\var{top[0]}}$</span></td>\n<td>$-$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>$\\color{green}0$</td>\n<td>.</td>\n<td>$\\color{green}{\\var{bot[2]}}$</td>\n<td>$\\var{bot[1]}$</td>\n<td>$\\var{bot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td>$\\color{red}0$</td>\n<td>.</td>\n<td>$\\color{red}{\\var{anshun}}$</td>\n<td>$\\var{ansten}$</td>\n<td>$\\var{ansunit}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{ans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: Multiplying and dividing by powers of ten", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>By powers of ten I mean a 1 followed by some 0s. The scientific notation questions will take care of the power of ten notation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on <em>Show steps</em> to see the full working. Then, once you understand how to do the question, click on <em>Try another question like this one</em> to start again.</p>", "advice": "<p>Multiplying or dividing by a power of ten (such as a $1$ followed by some $0$s) moves the decimal point. Multiplying moves the decimal point to make the number bigger (that is, to the right). Dividing moves the decimal to make the number smaller (that is, to the left). The number of $0$s indicates the number of places you should move the decimal place.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ans4": {"name": "ans4", "group": "Ungrouped variables", "definition": "dec4/poweroften[3]", "description": "", "templateType": "anything", "can_override": false}, "dec4": {"name": "dec4", "group": "Ungrouped variables", "definition": "random(list(0.111..0.499#0.001))", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "dec3/poweroften[2]", "description": "<p>ans3</p>", "templateType": "anything", "can_override": false}, "dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(list(0.111..0.999#0.001))*10", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "dec1*poweroften[0]", "description": "", "templateType": "anything", "can_override": false}, "poweroften": {"name": "poweroften", "group": "Ungrouped variables", "definition": "map(10^n,n,power)", "description": "", "templateType": "anything", "can_override": false}, "power": {"name": "power", "group": "Ungrouped variables", "definition": "shuffle([1,2,3,4])", "description": "", "templateType": "anything", "can_override": false}, "pronpower": {"name": "pronpower", "group": "Ungrouped variables", "definition": "[switch(power[0]=2,'two',power[0]=3,'three',power[0]=4,'four','one'),switch(power[1]=2,'two',power[1]=3,'three',power[1]=4,'four','one'),switch(power[2]=2,'two',power[2]=3,'three',power[2]=4,'four','one'),switch(power[3]=2,'two',power[3]=3,'three',power[3]=4,'four','one')]", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(list(0.011..0.099#0.0001))", "description": "", "templateType": "anything", "can_override": false}, "dec3": {"name": "dec3", "group": "Ungrouped variables", "definition": "random(list(0.500..0.999#0.001))*10", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "dec2*poweroften[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec1", "dec2", "dec3", "dec4", "power", "poweroften", "ans1", "ans2", "ans3", "ans4", "pronpower"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec1}\\times \\var{poweroften[0]}=$&nbsp;[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{poweroften[0]}$ has {pronpower[0]} $0$<span data-jme-visible=\"pronpower[0]&lt;&gt;1\">s</span> after the leading $1$. This means&nbsp;to&nbsp;evaluate $\\var{dec1}\\times \\var{poweroften[0]}$ we just move the decimal point in $\\var{dec1}$&nbsp;{pronpower[0]} decimal places to the right (to make the decimal $\\var{poweroften[0]}$ times bigger) and&nbsp;get $\\var{ans1}$.</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans1", "maxValue": "ans1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec2}\\times\\var{poweroften[1]}=$&nbsp;[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{poweroften[1]}$ has {pronpower[1]}&nbsp;$0$<span data-jme-visible=\"pronpower[1]&lt;&gt;1\">s</span> after the leading $1$. This means&nbsp;to&nbsp;evaluate $\\var{dec2}\\times \\var{poweroften[1]}$ we just move the decimal point in $\\var{dec2}$&nbsp;{pronpower[1]} decimal places to the right (to make the decimal $\\var{poweroften[1]}$ times bigger) and&nbsp;get $\\var{ans2}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans2", "maxValue": "ans2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec3}\\div\\var{poweroften[2]}=$&nbsp;[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{poweroften[2]}$ has {pronpower[2]}&nbsp;$0$<span data-jme-visible=\"pronpower[2]&lt;&gt;1\">s</span> after the leading $1$. This means&nbsp;to&nbsp;evaluate $\\var{dec3}\\div \\var{poweroften[2]}$ we just move the decimal point in $\\var{dec3}$&nbsp;{pronpower[2]} decimal places to the left (to make the decimal smaller) and&nbsp;get $\\var{ans3}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans3", "maxValue": "ans3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle \\frac{\\var{dec4}}{\\var{poweroften[3]}}=$&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Recall, the fraction bar simply denotes division.</p>\n<p>$\\var{poweroften[3]}$ has {pronpower[3]}&nbsp;$0$<span data-jme-visible=\"pronpower[3]&lt;&gt;1\">s</span> after the leading $1$. This means&nbsp;to&nbsp;evaluate $\\var{dec4}\\div \\var{poweroften[3]}$ we just move the decimal point in $\\var{dec4}$&nbsp;{pronpower[3]} decimal places to the left (to make the decimal smaller) and&nbsp;get $\\var{ans4}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans4", "maxValue": "ans4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: Multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>a) Multiplying decimals with a single non-zero digit. Students are told to preserve the number of decimal places (from the question to the answer).&nbsp;</p>\n<p>b) Multiplying decimals requiring the multiplication algorithm.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following questions down on paper and evaluate them without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ab0t0last": {"name": "ab0t0last", "group": "2digit", "definition": "mod(ab0t0,10)", "description": "", "templateType": "anything", "can_override": false}, "abot": {"name": "abot", "group": "2digit", "definition": "if(adigs[2]<>0,[adigs[3],adigs[2]],[adigs[2],adigs[3]])", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">abot</g></p>", "templateType": "anything", "can_override": false}, "ab1t0last": {"name": "ab1t0last", "group": "2digit", "definition": "mod(ab1t0,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t1carry": {"name": "ab0t1carry", "group": "2digit", "definition": "(ab0t1pluscarry-ab0t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "adps": {"name": "adps", "group": "2digit", "definition": "log(afactprod)", "description": "", "templateType": "anything", "can_override": false}, "adec1": {"name": "adec1", "group": "2digit", "definition": "atopnum/afact1", "description": "", "templateType": "anything", "can_override": false}, "afact1": {"name": "afact1", "group": "2digit", "definition": "random(10,100,1000,10000)", "description": "", "templateType": "anything", "can_override": false}, "adpsword": {"name": "adpsword", "group": "2digit", "definition": "switch(adps=6,\"six\", adps=5, \"five\", adps=4,\"four\", adps=3,\"three\", adps=2, \"two\", adps=1,\"one\",adps=7,\"seven\",adps)", "description": "", "templateType": "anything", "can_override": false}, "ab1t1carry": {"name": "ab1t1carry", "group": "2digit", "definition": "(ab1t1pluscarry-ab1t1last)/10", "description": "", "templateType": "anything", "can_override": false}, "adigs": {"name": "adigs", "group": "2digit", "definition": "shuffle(1..9)[0..4]", "description": "<p>we want distinct digits so it is easier to refer to digits unambiguously.&nbsp;</p>", "templateType": "anything", "can_override": false}, "ab0t1": {"name": "ab0t1", "group": "2digit", "definition": "abot[0]*atop[1]", "description": "", "templateType": "anything", "can_override": false}, "ab1t1": {"name": "ab1t1", "group": "2digit", "definition": "abot[1]*atop[1]", "description": "", "templateType": "anything", "can_override": false}, "ab0t1last": {"name": "ab0t1last", "group": "2digit", "definition": "mod(ab0t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "ab0t1pluscarry": {"name": "ab0t1pluscarry", "group": "2digit", "definition": "ab0t1+ab0t0carry", "description": "<p>ab</p>", "templateType": "anything", "can_override": false}, "aans": {"name": "aans", "group": "2digit", "definition": "atopnum*abotnum", "description": "", "templateType": "anything", "can_override": false}, "easydigprod": {"name": "easydigprod", "group": "Ungrouped variables", "definition": "easydig1*easydig2", "description": "", "templateType": "anything", "can_override": false}, "easyans": {"name": "easyans", "group": "Ungrouped variables", "definition": "easydigprod/(easyfactprod)", "description": "<p>eas</p>", "templateType": "anything", "can_override": false}, "easyfactprod": {"name": "easyfactprod", "group": "Ungrouped variables", "definition": "easyfact1*easyfact2", "description": "", "templateType": "anything", "can_override": false}, "easyfact2": {"name": "easyfact2", "group": "Ungrouped variables", "definition": "if(easyfact1=1,random(10,100,1000),random(1,10,100,1000))", "description": "", "templateType": "anything", "can_override": false}, "easy1": {"name": "easy1", "group": "Ungrouped variables", "definition": "easydig1/easyfact1", "description": "", "templateType": "anything", "can_override": false}, "afactprod": {"name": "afactprod", "group": "2digit", "definition": "afact1*afact2", "description": "", "templateType": "anything", "can_override": false}, "dps": {"name": "dps", "group": "Ungrouped variables", "definition": "log(easyfactprod)", "description": "", "templateType": "anything", "can_override": false}, "ab0t0carry": {"name": "ab0t0carry", "group": "2digit", "definition": "(ab0t0-ab0t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "aanstho": {"name": "aanstho", "group": "2digit", "definition": "mod((aans-aansone-aansten*10-aanshun*100)/1000,10)", "description": "", "templateType": "anything", "can_override": false}, "adec2": {"name": "adec2", "group": "2digit", "definition": "abotnum/afact2", "description": "", "templateType": "anything", "can_override": false}, "atopnum": {"name": "atopnum", "group": "2digit", "definition": "atop[1]*10+atop[0]", "description": "", "templateType": "anything", "can_override": false}, "ab1t1pluscarry": {"name": "ab1t1pluscarry", "group": "2digit", "definition": "ab1t1+ab1t0carry", "description": "", "templateType": "anything", "can_override": false}, "asum1": {"name": "asum1", "group": "2digit", "definition": "abot[0]*atopnum", "description": "", "templateType": "anything", "can_override": false}, "ab0t0": {"name": "ab0t0", "group": "2digit", "definition": "atop[0]*abot[0]", "description": "", "templateType": "anything", "can_override": false}, "easy2": {"name": "easy2", "group": "Ungrouped variables", "definition": "easydig2/easyfact2", "description": "", "templateType": "anything", "can_override": false}, "asum2": {"name": "asum2", "group": "2digit", "definition": "10*abot[1]*atopnum", "description": "<p>sum2</p>", "templateType": "anything", "can_override": false}, "aansten": {"name": "aansten", "group": "2digit", "definition": "mod((aans-aansone)/10,10)", "description": "", "templateType": "anything", "can_override": false}, "aansone": {"name": "aansone", "group": "2digit", "definition": "mod(aans,10)", "description": "", "templateType": "anything", "can_override": false}, "ab1t0carry": {"name": "ab1t0carry", "group": "2digit", "definition": "(ab1t0-ab1t0last)/10", "description": "", "templateType": "anything", "can_override": false}, "abotnum": {"name": "abotnum", "group": "2digit", "definition": "abot[1]*10+abot[0]", "description": "<p><g class=\"gr_ gr_3 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"3\" data-gr-id=\"3\">botnum</g></p>", "templateType": "anything", "can_override": false}, "atop": {"name": "atop", "group": "2digit", "definition": "if(adigs[0]<>0,[adigs[1],adigs[0]],[adigs[0],adigs[1]])", "description": "", "templateType": "anything", "can_override": false}, "ab1t0": {"name": "ab1t0", "group": "2digit", "definition": "abot[1]*atop[0]", "description": "", "templateType": "anything", "can_override": false}, "aanshun": {"name": "aanshun", "group": "2digit", "definition": "mod((aans-aansone-aansten*10)/100,10)", "description": "", "templateType": "anything", "can_override": false}, "adecans": {"name": "adecans", "group": "2digit", "definition": "aans/afactprod", "description": "", "templateType": "anything", "can_override": false}, "afact2": {"name": "afact2", "group": "2digit", "definition": "if(afact1=10000,random(10,100,1000),random(10,100,1000,10000))", "description": "", "templateType": "anything", "can_override": false}, "easyfact1": {"name": "easyfact1", "group": "Ungrouped variables", "definition": "random(1,10,100,1000)", "description": "", "templateType": "anything", "can_override": false}, "ab1t1last": {"name": "ab1t1last", "group": "2digit", "definition": "mod(ab1t1pluscarry,10)", "description": "", "templateType": "anything", "can_override": false}, "easydig1": {"name": "easydig1", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}, "easydig2": {"name": "easydig2", "group": "Ungrouped variables", "definition": "if(easydig1=3,random(4..9),random(3..9))", "description": "", "templateType": "anything", "can_override": false}, "dpsword": {"name": "dpsword", "group": "Ungrouped variables", "definition": "switch(dps=6,\"six\", dps=5, \"five\", dps=4,\"four\", dps=3,\"three\", dps=2, \"two\", dps=1,\"one\",dps)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["easydig1", "easydig2", "easyfact1", "easyfact2", "easydigprod", "easyfactprod", "easy1", "easy2", "easyans", "dps", "dpsword"], "variable_groups": [{"name": "2digit", "variables": ["adigs", "atop", "abot", "atopnum", "abotnum", "aans", "ab0t0", "ab0t0last", "ab0t0carry", "ab0t1", "ab0t1pluscarry", "ab0t1last", "ab0t1carry", "ab1t0", "ab1t0last", "ab1t0carry", "ab1t1", "ab1t1pluscarry", "ab1t1last", "ab1t1carry", "asum1", "asum2", "aansone", "aansten", "aanshun", "aanstho", "afact1", "afact2", "afactprod", "adps", "adpsword", "adec1", "adec2", "adecans"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{easy1}\\times \\var{easy2}= $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Remove the decimal points, do the multiplication of whole numbers, then put the decimal place in the answer so that the number of decimal places in the question and the answer are the same.</em></p>\n<p><em></em></p>\n<p>That is,</p>\n<ul>\n<li>since $\\var{easydig1}\\times\\var{easydig2}=\\var{easydigprod}$, the answer will involve the digit<span data-jme-visible=\"easydigprod&gt;9\">s</span> $\\var{easydigprod}$,</li>\n<li>since there <span data-jme-visible=\"dps&gt;=2\">are {dpsword} decimal places in the question, there will be {dpsword} decimal places in the answer,</span>&nbsp;<span data-jme-visible=\"dps=1\">is only one decimal place in the question, there will be one decimal place in the question,</span></li>\n</ul>\n<p><span data-jme-visible=\"mod(easydigprod,10)=0\">and therefore&nbsp;$\\var{easy1}\\times\\var{easy2}=\\var{easyans}0$. But note, we don't need to write the last zero so we could also write $\\var{easy1}\\times\\var{easy2}=\\var{easyans}$.</span></p>\n<p>and therefore&nbsp;$\\var{easy1}\\times\\var{easy2}=\\var{easyans}$.</p>\n<p></p>\n<p>This procedure works because it is&nbsp;the following in disguise:</p>\n<p>$\\begin{align}\\var{easy1}\\times\\var{easy2}&amp;=\\frac{\\var{easydig1}}{\\var{easyfact1}}\\times\\frac{\\var{easydig2}}{\\var{easyfact2}}&amp;&amp;\\text{(convert the decimals to fractions)}\\\\&amp;=\\frac{\\var{easydig1}\\times\\var{easydig2}}{\\var{easyfact1}\\times\\var{easyfact2}}&amp;&amp;\\text{(multiply the fractions)}\\\\&amp;=\\frac{\\var{easydigprod}}{\\var{easyfactprod}}\\\\&amp;=\\var{easyans}&amp;&amp;\\text{(convert back to a decimal)}\\end{align}$</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "easyans", "maxValue": "easyans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{adec1}\\times\\var{adec2} = $&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Remove the decimal points, do the multiplication of whole numbers, then put the decimal place in the answer so that the number of decimal places in the question and the answer are the same.</em></p>\n<p><em></em></p>\n<p>That is,</p>\n<ul>\n<li>since $\\var{atopnum}\\times\\var{abotnum}=\\var{aans}$ (see the working below), the answer will involve the digits $\\var{aans}$,</li>\n<li>since there <span data-jme-visible=\"adps&gt;=2\">are {adpsword} decimal places in the question, there will be {adpsword} decimal places in the answer,</span>&nbsp;<span data-jme-visible=\"adps=1\">is only one decimal place in the question, there will be one decimal place in the question,</span></li>\n</ul>\n<p><span data-jme-visible=\"mod(aans,10)=0 and mod(aans,100)&lt;&gt;0\">and therefore&nbsp;$\\var{adec1}\\times\\var{adec2}=\\var{adecans}0$. But note, we don't need to write the last zero so we could also write $\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.</span></p>\n<p><span data-jme-visible=\"mod(aans,100)=0\">and therefore&nbsp;$\\var{adec1}\\times\\var{adec2}=\\var{adecans}00$. But note, we don't need to write the two trailing zeros so we could also write $\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.</span></p>\n<p><span data-jme-visible=\"mod(aans,10)&lt;&gt;0\">and therefore&nbsp;$\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.</span></p>\n<p></p>\n<hr/>\n<p></p>\n<p>This procedure works because it is&nbsp;the following in disguise:</p>\n<p>$\\begin{align}\\var{adec1}\\times\\var{adec2}&amp;=\\frac{\\var{atopnum}}{\\var{afact1}}\\times\\frac{\\var{abotnum}}{\\var{afact2}}&amp;&amp;\\text{(convert the decimals to fractions)}\\\\&amp;=\\frac{\\var{atopnum}\\times\\var{abotnum}}{\\var{afact1}\\times\\var{afact2}}&amp;&amp;\\text{(multiply the fractions)}\\\\&amp;=\\frac{\\var{aans}}{\\var{afactprod}}\\\\&amp;=\\var{adecans}&amp;&amp;\\text{(convert back to a decimal)}\\end{align}$</p>\n<p></p>\n<hr/>\n<p></p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h3 style=\"text-align: center;\">How to calculate&nbsp;$\\var{atopnum}\\times\\var{abotnum}$</h3>\n<p>Generally&nbsp;we set up $\\var{atopnum}\\times\\var{abotnum}$&nbsp;with the <em>ones </em>and<em>&nbsp;</em><em>tens</em>&nbsp;columns lined up vertically:</p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\var{atop[1]}$</td>\n<td>$\\var{atop[0]}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\phantom{0}$</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>We need to multiply each digit in the bottom number by each digit in the top number whilst respecting their place values.</p>\n<p>&nbsp;</p>\n<p><strong>We multiply the digits in the ones column, that is, $\\color{green}{\\var{abot[0]}\\times \\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab0t0&lt;10\">Since this is just $\\var{ab0t0}$ we&nbsp;write $\\var{ab0t0}$ under the line in the ones column.</span></p>\n<p><span data-jme-visible=\"ab0t0&gt;=10\">Since this is $\\var{ab0t0}$ we write the $\\var{ab0t0last}$ under the line in the ones column and carry the&nbsp;$\\var{ab0t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab0t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}{\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p><strong>We now multiply&nbsp;diagonally, $\\color{green}{\\var{abot[0]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab0t1&lt;10 and ab0t0carry=0\">This just gives us $\\var{ab0t1}$ so we write&nbsp;$\\var{ab0t1}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t1&gt;=10 and ab0t0carry=0\">This gives us $\\var{ab0t1}$ so we write this under the line with the $\\var{ab0t1last}$ in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&gt;=10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1pluscarry}$ under the line with the $\\var{ab0t1last}$ in the tens column.</span></p>\n<p><span data-jme-visible=\"ab0t0carry&lt;&gt;0 and ab0t1pluscarry&lt;10\">This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab0t1pluscarry}$ under the line in the tens column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;&nbsp;</td>\n<td><span data-jme-visible=\"ab0t0carry&lt;&gt;0\">$\\color{green}{\\overset{{\\var{ab0t0carry}}}{\\var{atop[1]}}}$</span> <span data-jme-visible=\"ab0t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td>&nbsp;</td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\color{green}{\\var{abot[0]}}$</td>\n<td>&nbsp;</td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\color{red}{\\var{ab0t1carry}}$</span> <span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td>&nbsp;</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>We are now finished with the digit $\\var{abot[0]}$ and move on to work with the $\\var{abot[1]}$ in the tens column. Since this is really a $\\var{abot[1]*10}$ <strong>we place a zero in the ones column on the next line</strong> to pad our numbers out. <span data-jme-visible=\"ab0t0carry&lt;&gt;0\">We also crossout or erase any carry marks that we have used.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td>$\\overset{\\phantom{1}}{\\var{atop[1]}}$</td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td></td>\n<td>$\\color{red}0$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply along the other diagonal, that is, $\\color{green}{\\var{abot[1]}\\times\\var{atop[0]}}$.</strong></p>\n<p><span data-jme-visible=\"ab1t0&lt;10\">Since this is just $\\var{ab1t0}$ we&nbsp;write $\\var{ab1t0}$ under the line in the tens column.</span></p>\n<p><span data-jme-visible=\"ab1t0&gt;=10\">Since this is $\\var{ab1t0}$ we write the $\\var{ab1t0last}$ under the line in the tens column and carry the&nbsp;$\\var{ab1t0carry}$ into the tens column to be dealt with in the next step.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{\\color{red}{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[0]}}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td>$\\color{red}{\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now multiply the digits in the tens column, that is, $\\color{green}{\\var{abot[1]}\\times \\var{atop[1]}}$.&nbsp;</strong></p>\n<p><span data-jme-visible=\"ab1t1&lt;10 and ab1t0carry=0\">This just gives us $\\var{ab1t1}$ so we write&nbsp;$\\var{ab1t1}$ under the line in the&nbsp;hundreds column.</span></p>\n<p><span data-jme-visible=\"ab1t1&gt;=10 and ab1t0carry=0\">This gives us $\\var{ab1t1}$ so we write this under the line with the $\\var{ab1t1last}$ in the&nbsp;hundreds&nbsp;column.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&gt;=10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1pluscarry}$ under the line with the $\\var{ab1t1last}$ in the hundreds column.</span></p>\n<p><span data-jme-visible=\"ab1t0carry&lt;&gt;0 and ab1t1pluscarry&lt;10\">This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write&nbsp;$\\var{ab1t1pluscarry}$ under the line in the&nbsp;hundreds column.</span></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\color{green}{\\overset{{\\var{ab1t0carry}}}{\\var{atop[1]}}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\color{green}{\\overset{\\phantom{1}}{\\var{atop[1]}}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\color{green}{\\var{abot[1]}}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">${\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>${\\var{ab0t1last}}$</td>\n<td>${\\var{ab0t0last}}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\color{red}{\\var{ab1t1carry}}$</span> <span data-jme-visible=\"ab1t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{ab1t1last}}$</td>\n<td>${\\var{ab1t0last}}$</td>\n<td>${0}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>We now add the two results to get the total, that is, $\\color{green}{\\var{asum1}+\\var{asum2}}$.</strong></p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td></td>\n<td></td>\n<td><span data-jme-visible=\"ab1t0carry&lt;&gt;0\">$\\overset{{\\var{ab1t0carry}}}{\\var{atop[1]}}$</span> <span data-jme-visible=\"ab1t0carry=0\">$\\overset{\\phantom{1}}{\\var{atop[1]}}$</span></td>\n<td>&nbsp;$\\overset{\\phantom{1}}{\\var{atop[0]}}$</td>\n<td>$\\times$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td></td>\n<td></td>\n<td>$\\var{abot[1]}$</td>\n<td>$\\var{abot[0]}$</td>\n<td></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"ab0t1carry&lt;&gt;0\">$\\color{green}{\\var{ab0t1carry}}$</span><span data-jme-visible=\"ab0t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab0t1last}}$</td>\n<td>$\\color{green}{\\var{ab0t0last}}$</td>\n<td>$+$</td>\n</tr>\n<tr style=\"border-bottom: 1px solid black;\">\n<td><span data-jme-visible=\"ab1t1carry&lt;&gt;0\">$\\color{green}{\\var{ab1t1carry}}$</span> <span data-jme-visible=\"ab1t1carry=0\">$\\phantom{0}$</span></td>\n<td>$\\color{green}{\\var{ab1t1last}}$</td>\n<td>$\\color{green}{\\var{ab1t0last}}$</td>\n<td>$\\color{green}{0}$</td>\n<td></td>\n</tr>\n<tr>\n<td><span data-jme-visible=\"aanstho&lt;&gt;0\">$\\color{red}{\\var{aanstho}}$</span><span data-jme-visible=\"aanstho=0\">$\\phantom{0}$</span></td>\n<td>$\\color{red}{\\var{aanshun}}$</td>\n<td>$\\color{red}{\\var{aansten}}$</td>\n<td>$\\color{red}{\\var{aansone}}$</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p>The answer is therefore $\\var{aans}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "adecans", "maxValue": "adecans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Decimals: division (includes rounding the answer) - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Issues: alignment in columns in the working - not sure what to do about it</p>\n<p></p>\n<p>Decimal divided by a decimal. Multiply by a power of ten to get an integer divisor. Long and short division process. There is a remainder which we express as a decimal by continuing the division process. Rounding is required to some number of decimal places.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following question down on paper and evaluate&nbsp;it&nbsp;without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*1000),10)", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "0", "description": "<p>Was&nbsp;mod(floor(dividend1*100),10) but is now hard coded because there were rounding errors</p>", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..32 except [5,10,20,30])", "description": "<p>excluded 5 so that the decimal part is longer than 1 place.</p>", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "random(max(divisor1,25)..99 except list(divisor1/2..99#divisor1/2))/10\n//divisor1*quotient1+remainder/100", "description": "", "templateType": "anything", "can_override": false}, "divisorscaleorder": {"name": "divisorscaleorder", "group": "Ungrouped variables", "definition": "random(1,2,3,4)", "description": "", "templateType": "anything", "can_override": false}, "givendivisor": {"name": "givendivisor", "group": "Ungrouped variables", "definition": "divisor1/divisorscale", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything", "can_override": false}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "dividend1/divisor1\n//random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "description": "", "templateType": "anything", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", 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"diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(quotient1,2)", "description": "", "templateType": "anything", "can_override": false}, "givendividend": {"name": "givendividend", "group": "Ungrouped variables", "definition": "dividend1/divisorscale", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{givendividend}\\div\\var{givendivisor}=$[[0]]&nbsp;(2 decimal places)</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>We want to calculate $\\var{givendividend}\\div\\var{givendivisor}$.&nbsp;By the way,&nbsp;this is the same thing as $\\frac{\\var{givendividend}}{\\var{givendivisor}}$. Both of these expressions mean \"how many $\\var{givendivisor}$s go into $\\var{givendividend}$?\"</p>\n<p></p>\n<p>To work out such a thing we normally convert the division/fraction into an equivalent&nbsp;division/fraction where we are dividing by a whole number. We do this by multiplying both $\\var{givendividend}$ (the <em>dividend</em>) and $\\var{givendivisor}$ (the <em>divisor</em>) by $\\var{divisorscale}$ so that the decimal points are moved $\\var{divisorscaleorder}$ places to the right and so we get the division&nbsp;$\\var{dividend1}\\div\\var{divisor1}$.&nbsp;</p>\n<p><strong>Note</strong> we want the divisor to be a whole number but we don't need the dividend to be whole.</p>\n<p><strong>Why is this division equivalent?</strong> Think of division as sharing&nbsp;some amount&nbsp;equally amoungst some number of people.&nbsp;Now consider the scenario where you have&nbsp;$\\var{divisorscale}$ times the original amount to share but also&nbsp;$\\var{divisorscale}$ times the original number people, everyone will get the same amount as in the original scenario!&nbsp;</p>\n<p></p>\n<p></p>\n<p>The following is the above working in terms of fractions:</p>\n<p>\\begin{align}\\var{givendividend}\\div\\var{givendivisor}&amp;=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\\\[0.1cm]&amp;=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\times 1\\\\[0.1cm]&amp;=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\times\\frac{\\var{divisorscale}}{\\var{divisorscale}}\\\\[0.1cm]&amp;=\\frac{\\var{givendividend}\\times \\var{divisorscale}}{\\var{givendivisor}\\times\\var{divisorscale}}\\\\[0.1cm]&amp;=\\frac{\\var{dividend1}}{\\var{divisor1}}\\end{align}</p>\n<p></p>\n<p></p>\n<p></p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Short Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. By the way,&nbsp;this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The short division algorithm allows you to work this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!&nbsp;</p>\n<p>Actually, since we want the answer to<strong> two decimal places</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have&nbsp;<strong>three&nbsp;decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}}$</p>\n<p>Why three? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The ones column</strong></p>\n<p><strong>Step 1:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\"</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the ones column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the ones column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\,.\\phantom{^\\var{diff3}\\var{qd2}}\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\,.^\\phantom{\\var{diff3}}\\var{dd2}\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$</p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff3&lt;&gt;0\">We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the <em>remainder</em> (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the tenths column.</span> <span data-jme-visible=\"diff3=0\">We made it to exactly $\\var{dd3}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the tenths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this $\\var{b2}$ actually represents $\\var{b2*0.1}$)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the tenths column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the tenths column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff2&lt;&gt;0\">We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the <em>remainder</em> (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the hundredths column.</span> <span data-jme-visible=\"diff2=0\">We made it to exactly $\\var{b2}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the hundredths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The hundredths column</strong></p>\n<p><strong>Step 1: </strong>Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.01}$)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the hundredths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the hundredths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p><strong>Step 2: </strong><span data-jme-visible=\"diff1&lt;&gt;0\">We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the <em>remainder</em> (what remains to be divided) in the hundredths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the thousandths column.</span> <span data-jme-visible=\"diff1=0\">We made it to exactly $\\var{b1}$, so there is no remainder in the hundredths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the thousandths column. <em>You</em> do not need to do this but you can if you wish.</span></p>\n<p></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The thousandths column</strong></p>\n<p><strong>Step 1:</strong> Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.001}$)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the thousandths column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the thousandths column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the thousandths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}\\,\\,.\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$</p>\n<p><strong>Step 2:&nbsp;</strong>We could work out the remainder, we could keep adding zeros and continue the procedure but we only needed to determine the third decimal place in order to correctly round to the second decimal place and so we now stop the procedure.</p>\n<p>Since the third decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span><span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (2 dec. pl.).</p>\n<p></p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<hr/>\n<h2>Long Division</h2>\n<p>We want to calculate&nbsp;$\\var{dividend1}\\div\\var{divisor1}$. Note, this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$&nbsp;and both expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"</p>\n<p></p>\n<p>The long division algorithm allows you to work&nbsp;this out by working from the left to the right of $\\var{dividend1}$ whilst respecting place value. We normally set up the division in the following way:</p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$</p>\n<p>Note the positions of the numbers!</p>\n<p>Actually, since we want the answer to<strong> two decimal places</strong>&nbsp;we add as many zeroes after the decimal place to ensure we have&nbsp;<strong>three&nbsp;decimal places!&nbsp;</strong></p>\n<p style=\"text-align: left;\">$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}}$</p>\n<p>Why three? We use&nbsp;that extra digit to determine whether to round up or down.</p>\n<p>The algorithm (or procedure) seems complicated at first but you might find a&nbsp;mnemonic helps to remember the steps. We work left to right doing the following steps</p>\n<ol>\n<li>Divide</li>\n<li>Multiply</li>\n<li>Subtract</li>\n<li>Bring down&nbsp;</li>\n</ol>\n<p>and repeating until we run out of digits. The steps form the acronym <strong>DMSB</strong>. Popular&nbsp;mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\".</p>\n<p>&nbsp;</p>\n<p>We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.</p>\n<p>\\[\\boxed{\\begin{align}1\\times\\var{divisor1}&amp;=\\var{divisor1}\\\\2\\times\\var{divisor1}&amp;=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&amp;=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&amp;=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&amp;=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&amp;=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&amp;=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&amp;=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&amp;=\\var{9*divisor1}\\end{align}}\\]</p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;ones column</strong></p>\n<p><strong>D:</strong> The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\"&nbsp;</p>\n<p><span data-jme-visible=\"qd3=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the ones column:</span> <span data-jme-visible=\"qd3&gt;1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the&nbsp;ones column:&nbsp;</span> <span data-jme-visible=\"qd3=1\">Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd3}}$ above&nbsp;the $\\var{dd3}$ in the ones column:&nbsp;</span></p>\n<p>$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write&nbsp;$\\color{red}{\\var{prod3}}$ underneath in the&nbsp;ones column:</p>\n<p>&nbsp;$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{.000}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the <em>remainder</em> (what remains to be divided) in the ones column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{\\color{green}{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{.000}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd2}}$ in the&nbsp;tenths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff3}\\var{dd2}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}}\\phantom{.}\\color{red}{\\var{dd2}}\\phantom{00}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The tenths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this&nbsp;$\\var{b2}$ actually represents $\\var{b2/10}$ since it is in the&nbsp;tenths column)</p>\n<p><span data-jme-visible=\"qd2=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the&nbsp;tenths column:</span> <span data-jme-visible=\"qd2&gt;1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;tenths column:&nbsp;</span> <span data-jme-visible=\"qd2=1\">Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd2}}$ above&nbsp;the $\\var{dd2}$ in the&nbsp;tenths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{green}{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write&nbsp;$\\color{red}{\\var{prod2}}$ underneath in the tenths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{00}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the <em>remainder</em> (what remains to be divided) in the tenths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{green}{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{00}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd1}}$ in the hundredths column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff2}\\var{dd1}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{0}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The&nbsp;hundredths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this&nbsp;$\\var{b1}$ actually represents&nbsp;$\\var{b1/100}$ since it is in the hundredths column)</p>\n<p><span data-jme-visible=\"qd1=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the&nbsp;hundredths column:</span> <span data-jme-visible=\"qd1&gt;1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the&nbsp;hundredths column:&nbsp;</span> <span data-jme-visible=\"qd1=1\">Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd1}}$ above&nbsp;the $\\var{dd1}$ in the&nbsp;hundredths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{0}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write&nbsp;$\\color{red}{\\var{prod1}}$ underneath in the hundredths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{0}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the <em>remainder</em> (what remains to be divided) in the hundredths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{0}\\end{array}$</p>\n<p><strong>B:</strong> Now we bring the $\\color{green}{\\var{dd0}}$ in the&nbsp;thousandths&nbsp;column down next to the&nbsp;remainder so that&nbsp;it forms $\\var{diff1}\\var{dd0}$.</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>The thousandths column</strong></p>\n<p><strong>D:</strong>&nbsp;Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\"&nbsp;(note this&nbsp;$\\var{b0}$ actually represents&nbsp;$\\var{b0/1000}$ since it is in the thousandths column)</p>\n<p><span data-jme-visible=\"qd0=0\">Well, none!&nbsp;$\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the thousandths&nbsp;column:</span> <span data-jme-visible=\"qd0&gt;1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;thousandths&nbsp;column:&nbsp;</span> <span data-jme-visible=\"qd0=1\">Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits&nbsp;and we write&nbsp;$\\color{red}{\\var{qd0}}$ above&nbsp;the $\\var{dd0}$ in the&nbsp;thousandths column:&nbsp;</span></p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$</p>\n<p><strong>M:</strong> Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write&nbsp;$\\color{red}{\\var{prod0}}$ underneath in the thousandths column:</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$</p>\n<p><strong>S:</strong> We now do&nbsp;the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the <em>remainder</em> (what remains to be divided) in the thousandths column.&nbsp;</p>\n<p>$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm]&nbsp;\\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$</p>\n<p><br/><strong></strong></p>\n<p>Now we could keep adding zeros and continue the procedure but we only needed&nbsp;to determine&nbsp;the&nbsp;third decimal place in order to correctly round to&nbsp;two&nbsp;decimal places and so we now stop the procedure.</p>\n<p>Since the&nbsp;third decimal place was $\\var{qd0}$ we round <span data-jme-visible=\"qd0&gt;=5\">up</span>&nbsp;<span data-jme-visible=\"qd0&lt;5\">down</span> to $\\var{ans}$.&nbsp;Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (2 dec. pl.).</p>"}], "gaps": [{"type": "numberentry", 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{"name": "Merryn's copy of Decimals: rounding", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["decimals", "Decimals", "rounding"], "metadata": {"description": "<p>I think I prefer the other question I made called \"Rounding to 0, 1, 2 and 3 decimal places\"</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"wholelist": {"name": "wholelist", "group": "Ungrouped variables", "definition": "shuffle([whole1, whole2, whole3])", "description": "", "templateType": "anything", "can_override": false}, "whole1": {"name": "whole1", "group": "Ungrouped variables", "definition": "random(2..187)+random(list(0.0..0.4#0.1))+random(list(0.05..0.09#0.01))", "description": "", "templateType": "anything", "can_override": false}, "whole3": {"name": "whole3", "group": "Ungrouped variables", "definition": "random(2..187)+0.5", "description": "", "templateType": "anything", "can_override": false}, "twolist": {"name": "twolist", "group": "Ungrouped variables", "definition": "shuffle([two1,two2,two3])", "description": "", "templateType": "anything", "can_override": false}, "two3": {"name": "two3", "group": "Ungrouped variables", "definition": "random(2..187)+random(list(0.0..0.9#0.1))+random(list(0.00..0.04#0.01))+0.005+random(list(0.00001..0.00009#0.00001))", "description": "", "templateType": "anything", "can_override": false}, "whole2": {"name": "whole2", "group": "Ungrouped variables", "definition": 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false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Round the following decimals to&nbsp;the nearest whole number ($0$ decimal places):</p>\n<p>$\\var{wholelist[0]}=$ [[0]]</p>\n<p>$\\var{wholelist[1]}=$ [[1]]</p>\n<p>$\\var{wholelist[2]}=$ [[2]]</p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>If it is 5 or more, round up, otherwise round down.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>To round to a whole number you look at the first decimal place to determine whether you round up or down. If the first decimal place is a 5 or more then you round up (increase the units column by 1) and discard the decimals. If the first decimal 4 or less then you round down (leave the units column as it is) and discard the decimals.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example:</p>\n<ul>\n<li>123.49 is&nbsp;123 to the nearest whole number</li>\n<li>123.5 is 124 to the nearest whole number</li>\n</ul>\n<p></p>\n<hr/>\n<p></p>\n<p>Note we only look at the first decimal place when rounding to whole numbers, the others are irrelevant, that is 123.4998 should still be rounded to 123. &nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(wholelist[0],0)", "maxValue": "precround(wholelist[0],0)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(wholelist[1],0)", "maxValue": "precround(wholelist[1],0)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(wholelist[2],0)", "maxValue": "precround(wholelist[2],0)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Round the following decimals to $2$ decimal places:</p>\n<p>$\\var{twolist[0]}=$ [[0]]</p>\n<p>$\\var{twolist[1]}=$ [[1]]</p>\n<p>$\\var{twolist[2]}=$ [[2]]</p>\n<p><em>Note: Suppose your answer was $123.4$, you need to enter your answer as $123.40$ since the question asks for $2$ decimal places.</em></p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>If it is 5 or more, round up, otherwise round down.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>Since&nbsp;we are told to round to two&nbsp;decimal places we look at the third decimal place to determine if we round up or down.&nbsp;If the third&nbsp;decimal place is a 5 or more then you round up (increase the second decimal place&nbsp;by 1) and discard the digits to the right. If the third&nbsp;decimal place is 4 or less then you round down (leave the second decimal place&nbsp;as it is) and discard the digits to the right.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example:</p>\n<ul>\n<li>123.3461 is&nbsp;123.35 to&nbsp;2 decimal places</li>\n<li>123.5735 is 123.57&nbsp;to&nbsp;2 decimal places</li>\n<li>123.995 is 124.00 to 2 decimal places&nbsp;</li>\n</ul>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(twolist[0],2)", "maxValue": "precround(twolist[0],2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": 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your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "precround(twolist[2],2)", "maxValue": "precround(twolist[2],2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": 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"https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}, {"name": "Laura Midgley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18287/"}], "tags": ["category: numbers", "leads to: 107, Ratios", "leads to: 206, Cancelling numerical fractions", "leads to: 208, Factors of algebraic products", "skill: 108, Factors of an integer"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"scenarios": {"name": "scenarios", "group": "Ungrouped variables", "definition": "[\n  [\n    [\n      \"15 and 25 are both factors of 75\", \n      \"8 is a factor of 100\",\n      \"6, 8 and 16 are all factors of 48\",\n      \"5, 8 and 12 are all factors of 60\",\n    ],\n    [1,0,1,0]\n  ],\n  [\n    [\n      \"6 and 9 are both factors of 96\",\n      \"6, 16 and 24 are all factors of 64\",\n      \"3, 8 and 18 are all factors of 72\",\n      \"35 is a factor of 105\"\n    ],\n    [0,0,1,1]\n  ]\n]", "description": "", "templateType": "anything", "can_override": false}, "scenario": {"name": "scenario", "group": "Ungrouped variables", "definition": "random(scenarios)", "description": "", "templateType": "anything", "can_override": false}, "choices": {"name": "choices", "group": "Ungrouped variables", "definition": "scenario[0]", "description": "", "templateType": "anything", "can_override": false}, "correct_matrix": {"name": "correct_matrix", "group": "Ungrouped variables", "definition": "scenario[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["scenarios", "scenario", "choices", "correct_matrix"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which of the following are true?</p>", "minMarks": 0, "maxMarks": "1", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "2", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "score per matched cell", "choices": "choices", "matrix": "correct_matrix"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Primes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Simple questions on square numbers, square roots, cube number, cube roots and prime numbers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Answer the following.</p>", "advice": "<p>A&nbsp;number ($x$) squared ($x^2$) is that number times itself ($x^2$ =&nbsp;$x$ x&nbsp;$x$).</p>\n<p>A&nbsp;number ($x$) cubed&nbsp;($x^3$) is that number times itself ties itself ($x^3$ =&nbsp;$x$ x&nbsp;$x$ x&nbsp;$x$).</p>\n<p>A square root ($\\sqrt{x}$) of a number$x$, is the number that you have to square in order to get&nbsp;$x$.</p>\n<p>A cube root ($\\sqrt[3]{x}$) of a number$x$, is the number that you have to cube in order to get&nbsp;$x$.</p>\n<p>Prime numbers are numbers with only two factors - 1 and themself.&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"g": {"name": "g", "group": "Ungrouped variables", "definition": "random(11,13,17,19,23)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2,3,5,7)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(0,1,4,6,8,9,10,12,14)", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(15,16,18,20,21,22,24,25)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f", "g", "h", "j"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which two of the following are prime numbers?</p>", "minMarks": 0, "maxMarks": "2", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$\\var{f}$</p>", "<p>$\\var{g}$</p>", "<p>$\\var{h}$</p>", "<p>$\\var{j}$</p>"], "matrix": ["1", "1", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of LCM and HCF", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Calculating the LCM and HCF of numbers by using prime factorisation.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>$x = \\var{x}$</p>\n<p>$y = \\var{y}$</p>\n<p>$z = \\var{z}$</p>\n<p></p>\n<p>By considering the prime factorisation of $x, y $ and $z$, or otherwise, find:</p>", "advice": "<p>We can write $x,y,z$ as a product of prime factors as follows:</p>\n<p>$x=\\var{x}=\\var{show_factors(x)}$</p>\n<p>$y=\\var{y}=\\var{show_factors(y)}$</p>\n<p>$z=\\var{z}=\\var{show_factors(z)}$</p>\n<p></p>\n<p>For&nbsp;HCF of $\\var{x}$ and $\\var{y}$ we need to multiply each factor the least number of times it occurs in either $\\var{x}$ or $\\var{y}$</p>\n<p>i.e. HCF$(x,y) = \\var{show_factors(hcf_xy)}=\\var{hcf_xy}$</p>\n<p></p>\n<p>For&nbsp;LCM of $\\var{x}$ and $\\var{y}$ we need to multiply each factor the greatest number of times it occurs in either $\\var{x}$ or $\\var{y}$</p>\n<p>i.e. LCM$(x,y) = \\var{show_factors(lcm_xy)}=\\var{lcm_xy}$</p>\n<p></p>\n<p>For&nbsp;HCF of $\\var{x}$ and $\\var{z}$ we need to multiply each factor the least number of times it occurs in either $\\var{x}$ or $\\var{z}$</p>\n<p>i.e. HCF$(x,z) = \\var{show_factors(hcf_xz)}=\\var{hcf_xz}$</p>\n<p></p>\n<p>For&nbsp;LCM of $\\var{x}$ and $\\var{z}$ we need to multiply each factor the&nbsp;greatest&nbsp;number of times it occurs in either $\\var{x}$ or $\\var{z}$</p>\n<p>i.e. LCM$(x,z) = \\var{show_factors(lcm_xz)}=\\var{lcm_xz}$</p>\n<p></p>\n<p>For HCF of $\\var{x},\\var{y}$ and $\\var{z}$ we need to multiply each factor the least number of times it occurs in either $\\var{x}$ or $\\var{y}$&nbsp;or $\\var{z}$</p>\n<p>i.e. HCF$(x,y,z) = \\var{show_factors(hcf_xyz)}=\\var{hcf_xyz}$</p>\n<p></p>\n<p>For&nbsp;LCM of $\\var{x},\\var{y}$ and $\\var{z}$ we need to multiply each factor the&nbsp;greatest&nbsp;number of times it occurs in either $\\var{x}$ or $\\var{y}$ or $\\var{z}$</p>\n<p>i.e. LCM$(x,y,z) = \\var{show_factors(lcm_xyz)}=\\var{lcm_xyz}$</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x_powers": {"name": "x_powers", "group": "Ungrouped variables", "definition": "[random(1..4),random(0..4),random(0..3),random(0..3)]", "description": "", "templateType": "anything", "can_override": false}, "y_powers": {"name": "y_powers", "group": "Ungrouped variables", "definition": "[random(0..4),random(1..4),random(0..3),random(0..2)]", "description": "", "templateType": "anything", "can_override": false}, "z_powers": {"name": "z_powers", "group": "Ungrouped variables", "definition": "[random(1..4),random(1..4),random(0..3),random(1..2)]", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "2^x_powers[0]*3^x_powers[1]*5^x_powers[2]*7^x_powers[3]", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "2^y_powers[0]*3^y_powers[1]*5^y_powers[2]*7^y_powers[3]", "description": "", "templateType": "anything", "can_override": false}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "2^z_powers[0]*3^z_powers[1]*5^z_powers[2]*7^z_powers[3]", "description": "", "templateType": "anything", "can_override": false}, "hcf_xy": {"name": "hcf_xy", "group": "Ungrouped variables", "definition": "2^min(x_powers[0],y_powers[0])*3^min(x_powers[1],y_powers[1])*5^min(x_powers[2],y_powers[2])*7^min(x_powers[3],y_powers[3])", "description": "", "templateType": "anything", "can_override": false}, "hcf_xz": {"name": "hcf_xz", "group": "Ungrouped variables", "definition": "2^min(x_powers[0],z_powers[0])*3^min(x_powers[1],z_powers[1])*5^min(x_powers[2],z_powers[2])*7^min(x_powers[3],z_powers[3])", "description": "", "templateType": "anything", "can_override": false}, "lcm_xy": {"name": "lcm_xy", "group": "Ungrouped variables", "definition": "2^max(x_powers[0],y_powers[0])*3^max(x_powers[1],y_powers[1])*5^max(x_powers[2],y_powers[2])*7^max(x_powers[3],y_powers[3])", "description": "", "templateType": "anything", "can_override": false}, "lcm_xz": {"name": "lcm_xz", "group": "Ungrouped variables", "definition": "2^max(x_powers[0],z_powers[0])*3^max(x_powers[1],z_powers[1])*5^max(x_powers[2],z_powers[2])*7^max(x_powers[3],z_powers[3])", "description": "", "templateType": "anything", "can_override": false}, "hcf_xyz": {"name": "hcf_xyz", "group": "Ungrouped variables", "definition": "2^min([x_powers[0],y_powers[0],z_powers[0]])*3^min([x_powers[1],y_powers[1],z_powers[1]])*5^min([x_powers[2],y_powers[2],z_powers[2]])*7^min([x_powers[3],y_powers[3],z_powers[3]])", "description": "", "templateType": "anything", "can_override": false}, "lcm_xyz": {"name": "lcm_xyz", "group": "Ungrouped variables", "definition": "2^max([x_powers[0],y_powers[0],z_powers[0]])*3^max([x_powers[1],y_powers[1],z_powers[1]])*5^max([x_powers[2],y_powers[2],z_powers[2]])*7^max([x_powers[3],y_powers[3],z_powers[3]])", "description": "", "templateType": "anything", "can_override": false}, "primes": {"name": "primes", "group": "Ungrouped variables", "definition": "[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "x_powers[0]+x_powers[1]+x_powers[2]+x_powers[3]<8\nand\nx_powers[0]+x_powers[1]+x_powers[2]+x_powers[3]>1\nand\ny_powers[0]+y_powers[1]+y_powers[2]+y_powers[3]<8\nand\ny_powers[0]+y_powers[1]+y_powers[2]+y_powers[3]>1\nand\nz_powers[0]+z_powers[1]+z_powers[2]+z_powers[3]<8\nand\nz_powers[0]+z_powers[1]+z_powers[2]+z_powers[3]>1\nand (x-y)(y-z)(x-z)<>0\nand hcf_xyz>1", "maxRuns": "500"}, "ungrouped_variables": ["x_powers", "y_powers", "z_powers", "x", "y", "z", "hcf_xy", "hcf_xz", "lcm_xy", "lcm_xz", "hcf_xyz", "lcm_xyz", "primes"], "variable_groups": [], "functions": {"show_factors": {"parameters": [["n", "number"]], "type": "string", "language": "jme", "definition": "latex(  // mark the output as a string of raw LaTeX\n  join(\n    map(\n      if(a=1,p,p+'^{'+a+'}'), // when the exponent is 1, return p, otherwise return p^{exponent}\n      [p,a],\n      filter(x[1]>0,x,zip(primes,factorise(n)))  // for all the primes p which are factors of n, return p and its exponent\n    ),\n    ' \\\\times ' // join all the prime powers up with \\times symbols\n  )\n)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The highest common factor (HCF) of $x$ and $y$</p>", "minValue": "hcf_xy", "maxValue": "hcf_xy", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The&nbsp;lowest common multiple (LCM) of $x$ and $y$</p>", "minValue": "lcm_xy", "maxValue": "lcm_xy", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The HCF of $x$ and $z$</p>", "minValue": "hcf_xz", "maxValue": "hcf_xz", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The LCM of $x$ and $z$</p>", "minValue": "lcm_xz", "maxValue": "lcm_xz", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The HCF of $x,y$ and $z$</p>", "minValue": "hcf_xyz", "maxValue": "hcf_xyz", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The&nbsp;LCM&nbsp;of $x,y$ and $z$</p>", "minValue": "lcm_xyz", "maxValue": "lcm_xyz", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Fractions", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Fractions: equivalent", "Fractions: adding and subtracting, numerical, already with common denominator", "Fractions: adding and subtracting, numerical", "Fractions/division and multiplication, different ways of presenting the same thing (non-algebraic)", "Fractions: improper and mixed numerals", "Fractions: operations involving numerical fractions", "Fraction of a pizza remaining (4)", "Converting between Mixed Numbers and Improper Fractions", "Fractions: improper and mixed numerals", "Working with Mixed Numbers", " Division of fractions"], "variable_overrides": [[], [], [], [], [], [], [], [], [], [], []], "questions": [{"name": "Merryn's copy of Fractions: equivalent", "extensions": [], "custom_part_types": [], "resources": [["question-resources/equivalent_fractions.svg", "/srv/numbas/media/question-resources/equivalent_fractions.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["cancelling", "equal fractions", "equivalent fractions", "Fractions", "fractions", "lowest form", "reducing fractions", "simplest form", "simplifying"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {"std": ["all"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"denom1": {"name": "denom1", "group": "Ungrouped variables", "definition": "random(2..12 except num1)", "description": "", "templateType": "anything", "can_override": false}, "num4": {"name": "num4", "group": "Ungrouped variables", "definition": "num1*mult3", "description": "", "templateType": "anything", "can_override": false}, "denom3": {"name": "denom3", "group": "Ungrouped variables", "definition": "denom1*mult2", "description": "", "templateType": "anything", "can_override": false}, "denom2": {"name": "denom2", "group": "Ungrouped variables", "definition": "denom1*mult1", "description": "", "templateType": "anything", "can_override": false}, "denom5": {"name": "denom5", "group": "Ungrouped variables", "definition": "random(2..12 except num5)", "description": "", "templateType": "anything", "can_override": false}, "num1": {"name": "num1", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "num2": {"name": "num2", "group": "Ungrouped variables", "definition": "num1*mult1", "description": "", "templateType": "anything", "can_override": false}, "num3": {"name": "num3", "group": "Ungrouped variables", "definition": "num1*mult2", "description": "", "templateType": "anything", "can_override": false}, "denom6": {"name": "denom6", "group": "Ungrouped variables", "definition": "denom5*mult4", "description": "", "templateType": "anything", "can_override": false}, "num6": {"name": "num6", "group": "Ungrouped variables", "definition": "num5*mult4", "description": "", "templateType": "anything", "can_override": false}, "denom4": {"name": "denom4", "group": "Ungrouped variables", "definition": "denom1*mult3", "description": "", "templateType": "anything", "can_override": false}, "num5": {"name": "num5", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "mult3": {"name": "mult3", "group": "Ungrouped variables", "definition": "random(2..10 except [mult1, mult2])", "description": "", "templateType": "anything", "can_override": false}, "mult2": {"name": "mult2", "group": "Ungrouped variables", "definition": "random(2..10 except mult1)", "description": "", "templateType": "anything", "can_override": false}, "mult1": {"name": "mult1", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "mult4": {"name": "mult4", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "mult6": {"name": "mult6", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "mult5": {"name": "mult5", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "Name": {"name": "Name", "group": "Ungrouped variables", "definition": "repeat(random([\"Ben\", \"He\"], [\"Annie\", \"She\"], [\"Matt\", \"He\"], [\"David\", \"He\"], [\"Steve\", \"He\"], [\"David\", \"He\"], [\"Scott\", \"He\"], [\"Fran\", \"She\"], [\"Jenny\", \"She\"], [\"Lyn\", \"She\"], [\"Judy-anne\", \"She\"], [\"Courtney\", \"She\"]),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num1", "denom1", "mult1", "num2", "denom2", "mult2", "num3", "denom3", "mult3", "num4", "denom4", "num5", "denom5", "mult4", "num6", "denom6", "Name", "mult5", "mult6"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ".fractiontable table {\n  width: 40%; \n  padding: 0px; \n  border-width: 0px; \n  layout: fixed;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  width: 15%; \n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n.fractiontable .tdeq \n{\n  width: 5%; \n  border-bottom: 0px;\n  font-size: x-large;\n}\n\n\n.fractiontable th {\n    background-color:#aaa;\n}\n/*Fix the height of all cells EXCEPT table-headers to 40px*/\n.fractiontable td {\n    height:40px;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{Name[0][0]} has written $\\frac{\\var{num5}}{\\var{denom5}}$ in the equivalent form $\\frac{\\var{num6}}{\\var{denom6}}$.</p>\n<p></p>\n<p>What has {Name[0][0]} done to the first&nbsp;fraction in order to&nbsp;get the second? {Name[0][1]} has multiplied the top and bottom by&nbsp;[[0]]&nbsp;.</p>", "stepsPenalty": "0.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If we multiply the top and bottom of a fraction by a&nbsp;number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example, suppose you cut a cake up into 3 parts and throw away one piece, what is left is <g class=\"gr_ gr_20 gr-alert gr_spell ContextualSpelling multiReplace\" id=\"20\" data-gr-id=\"20\">two thirds</g> of the whole cake, that is, $\\frac{2}{3}$ of the whole cake. Now suppose you have another identical cake, this time you cut it into 6 parts and throw away&nbsp;two parts, what is left is four sixths of the whole cake, that is, $\\frac{4}{6}$ of the whole cake.</p>\n<p>Notice in both situations you end up with the same amount of cake!&nbsp;</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/equivalent_fractions.svg\"/></p>\n<p>So $\\frac{2}{3}$ is equivalent to $\\frac{4}{6}$ and we can write \\[\\frac{2}{3}=\\frac{4}{6}\\]</p>\n<p>If you look at the numbers you might notice that for the second cake we just doubled all the numbers, and in the second fraction all the numbers are two times those in the first fraction. In general equivalent fractions are formed by multiplying (or dividing) the top and bottom of a fraction by the same number.&nbsp;</p>\n<p>So if you were asked how a&nbsp;person got from $\\frac{5}{6}$ to the equivalent fraction $\\frac{20}{24}$ you ask yourself&nbsp;'what do I&nbsp;multiply 5 by to get 20?' and 'what do I multiply 6 by to get 24?' and then realise they must have done the following</p>\n<p>\\[\\frac{5}{6}=\\frac{5\\times 4}{6\\times 4}=\\frac{20}{24}\\]</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "mult4", "maxValue": "mult4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{Name[1][0]} has written $\\frac{\\var{num2*mult5}}{\\var{denom2*mult5}}$ in the equivalent form $\\frac{\\var{num2}}{\\var{denom2}}$.</p>\n<p></p>\n<p>What has {Name[1][0]} done to the first&nbsp;fraction in order to&nbsp;get the second? {Name[1][1]} has divided&nbsp;the top and bottom by&nbsp;[[0]]&nbsp;.</p>", "stepsPenalty": "0.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If we divide&nbsp;the top and bottom of a fraction by a&nbsp;number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example, suppose you cut it into 6 parts and throw away&nbsp;two parts, what is left is four sixths of the whole cake, that is, $\\frac{4}{6}$ of the whole cake. Now suppose you have another identical cake, this time you cut a cake up into 3 parts and throw away one piece, what is left is two thirds of the whole cake, that is, $\\frac{2}{3}$ of the whole cake.</p>\n<p>Notice in both situations you end up with the same amount of cake!</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/equivalent_fractions.svg\"/></p>\n<p>So $\\frac{4}{6}$ is equivalent to $\\frac{2}{3}$ and we can write \\[\\frac{4}{6}=\\frac{2}{3}\\]</p>\n<p>If you look at the numbers you might notice that for the second cake we just halved&nbsp;all the numbers, and in the second fraction all the numbers are half of&nbsp;those in the first fraction. In general equivalent fractions are formed by dividing&nbsp;(or multiplying) the top and bottom of a fraction by the same number.&nbsp;</p>\n<p>So if you were asked how a&nbsp;person got from $\\frac{20}{24}$ to the equivalent fraction $\\frac{5}{6}$ you ask yourself&nbsp;'what do I divide&nbsp;20 by to get 5?' and 'what do I divide 24&nbsp;by to get 6?' and then realise they must have done the following</p>\n<p>\\[\\frac{20}{24}=\\frac{20\\div 4}{24\\div 4}=\\frac{5}{6}\\]</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "mult5", "maxValue": "mult5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<div class=\"fractiontable\">\n<p>Please enter numbers to create equivalent fractions.</p>\n<table>\n<tbody>\n<tr>\n<td class=\"tdnum\">{num1}</td>\n<td class=\"tdeq\">=</td>\n<td class=\"tdnum\">[[0]]</td>\n<td class=\"tdeq\">=</td>\n<td class=\"tdnum\">{num3}</td>\n<td class=\"tdeq\">=</td>\n<td class=\"tdnum\">[[2]]</td>\n</tr>\n<tr>\n<td class=\"tddenom\">{denom1}</td>\n<td></td>\n<td class=\"tddenom\">{denom2}</td>\n<td></td>\n<td class=\"tddenom\">[[1]]</td>\n<td></td>\n<td class=\"tddenom\">{denom4}</td>\n</tr>\n</tbody>\n</table>\n</div>", "stepsPenalty": "1.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>These are equivalent fractions so the same number that multiplied the numerator must multiply the denominator.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example given:</p>\n<p>\\[\\frac{8}{5}=\\frac{}{15}\\]</p>\n<p>you can see the denominator of 5 was multiplied by 3 to become 15, so to make an equivalent fraction, we would need to multiply the numerator by 3&nbsp;as well. So the blank must be $8\\times 3$ which is $24$. Your working might look like this:</p>\n<p>\\[\\frac{8}{5}=\\frac{8\\times 3}{5\\times 3}=\\frac{24}{15}\\]&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num2", "maxValue": "num2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "denom3", "maxValue": "denom3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num4", "maxValue": "num4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<div class=\"fractiontable\">\n<p>Simplify the following fractions into their&nbsp;lowest forms:</p>\n<p>(for this question, if your fraction turns into a whole number write it over 1)</p>\n<table>\n<tbody>\n<tr>\n<td class=\"tdnum\">{num2*mult6}</td>\n<td class=\"tdeq\" style=\"text-align: center;\">=</td>\n<td class=\"tdnum\" style=\"text-align: center;\">[[0]]</td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n</tr>\n<tr>\n<td class=\"tddenom\">{denom4*mult6}</td>\n<td></td>\n<td class=\"tddenom\">[[1]]</td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td class=\"tdnum\">{num3*mult3}</td>\n<td class=\"tdeq\" style=\"text-align: center;\">=</td>\n<td class=\"tdnum\" style=\"text-align: center;\">[[2]]</td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n</tr>\n<tr>\n<td class=\"tddenom\">{denom1*mult3}</td>\n<td></td>\n<td class=\"tddenom\">[[3]]</td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n<td class=\"tdeq\"></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n</div>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Divide the top and bottom by their highest common factor, or repeatedly divide the top and bottom by common factors.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>We can write a fraction in a lower form by dividing the top and bottom by the same number. We can repeatedly do this until there are no numbers that 'go evenly into' both the top and the bottom. At this point, the number is in 'lowest form'.</p>\n<p></p>\n<p>For example:</p>\n<p>To simplify $\\frac{360}{132}$ we might first notice that we can divide both the top and bottom by 2 to get $\\frac{180}{66}$, then&nbsp;you might realise you can divide both the top and bottom by 6 to get $\\frac{30}{11}$. At this point, there is no number that will divide both evenly because they have no common factor other than 1. Your working could look like this:</p>\n<p>\\[\\frac{360}{132}=\\frac{360\\div 2}{132\\div 2}=\\frac{180}{66}=\\frac{180\\div 6}{66\\div 6}=\\frac{30}{11}\\]</p>\n<p>Notice dividing by 2 and then by 6 is the same as dividing by 12, in this example 12 is the highest common factor of the top and the bottom of the fraction.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>In general, if you can determine the highest common factor of the two numbers in the fraction you should then divide the top and bottom of the fraction by this number, then there will be no common factors left to divide by and the fraction will be in its lowest form.</p>\n<p></p>\n<p>For example:&nbsp;</p>\n<p>To simplify&nbsp;$\\frac{360}{132}$ you determine the highest common factor of $360$ and $132$ is $12$, and so divide the top and bottom of the fraction by $12$ to get $\\frac{30}{11}$. Your working could look like this:</p>\n<p>\\[\\frac{360}{132}=\\frac{360\\div 12}{132\\div 12}=\\frac{30}{11}\\]</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num2/gcd(num2,denom4)", "maxValue": "num2/gcd(num2,denom4)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "denom4/gcd(num2,denom4)", "maxValue": "denom4/gcd(num2,denom4)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num3/gcd(num3,denom1)", "maxValue": "num3/gcd(num3,denom1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "denom1/gcd(num3,denom1)", "maxValue": "denom1/gcd(num3,denom1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fractions: adding and subtracting, numerical, already with common denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Fractions already have a common denominator. Addition and subtraction 50:50 split, when subtracting, the answer is negative half the time. Students shouldn't have to worry about reducing fractions by design.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Express the following as a single fraction (using / as the fraction bar).</p>", "advice": "<p>Since both denominators are the same, i.e. $\\var{den}$. We can simply <span data-jme-visible=\"addSub=1\">add</span><span data-jme-visible=\"addSub=-1\">subtract</span> the numerators, $\\var{num1}$ and $\\var{num2}$, and put that over the common denominator.</p>\n<p><span data-jme-visible=\"addSub=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den}}+\\dfrac{\\var{num2}}{\\var{den}}&amp;=\\dfrac{\\var{num1}+\\var{num2}}{\\var{den}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{den}}\\end{align*}$</span></p>\n<p><span data-jme-visible=\"addSub=-1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den}}-\\dfrac{\\var{num2}}{\\var{den}}&amp;=\\dfrac{\\var{num1}-\\var{num2}}{\\var{den}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{den}}\\end{align*}$</span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"den": {"name": "den", "group": "Ungrouped variables", "definition": "numbers[2]", "description": "", "templateType": "anything", "can_override": false}, "num1": {"name": "num1", "group": "Ungrouped variables", "definition": "numbers[0]", "description": "", "templateType": "anything", "can_override": false}, "num2": {"name": "num2", "group": "Ungrouped variables", "definition": "numbers[1]", "description": "", "templateType": "anything", "can_override": false}, "addSub": {"name": "addSub", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "<p>adding or subtracting</p>", "templateType": "anything", "can_override": false}, "ansnum": {"name": "ansnum", "group": "Ungrouped variables", "definition": "if(addSub=1,num1+num2,num1-num2)", "description": "", "templateType": "anything", "can_override": false}, "numbers": {"name": "numbers", "group": "Ungrouped variables", "definition": "random(tempnumbers,[tempnumbers[1],tempnumbers[0],tempnumbers[2]])", "description": "", "templateType": "anything", "can_override": false}, "tempnumbers": {"name": "tempnumbers", "group": "Ungrouped variables", "definition": "sort(shuffle([3,5,7,11,13,17,19,23,29])[0..3])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["tempnumbers", "numbers", "den", "num1", "num2", "addSub", "ansnum"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><span data-jme-visible=\"addSub=1\">$\\dfrac{\\var{num1}}{\\var{den}}+\\dfrac{\\var{num2}}{\\var{den}}$</span> <span data-jme-visible=\"addSub=-1\">$\\dfrac{\\var{num1}}{\\var{den}}-\\dfrac{\\var{num2}}{\\var{den}}$</span>&nbsp; $=$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansNum/den", "maxValue": "ansNum/den", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fractions: adding and subtracting, numerical", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Fractions don't have a common denominator. Need to find one. Addition and subtraction 50:50 split.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Express the following as a single fraction (using / as the fraction bar).</p>", "advice": "<p>We need to get the denominators to be the same, preferably the lowest common denominator, at which point, we can simply <span data-jme-visible=\"addSub=1\">add</span><span data-jme-visible=\"addSub=-1\">subtract</span> the new numerators and put the result over the common denominator.</p>\n<div data-jme-visible=\"cf&lt;&gt;1 and extrafactor[0]&lt;&gt;1 and extrafactor[1]&lt;&gt;1\">\n<p>Since $\\var{den1}=\\var{cf}\\times\\var{extrafactor[0]}$ and $\\var{den2}=\\var{cf}\\times \\var{extrafactor[1]}$ the lowest common denominator will be the product $\\var{cf}\\times \\var{extrafactor[0]}\\times \\var{extrafactor[1]}$, that is, $\\var{ansden}$. Therefore, we will multiply the first fraction by $\\frac{\\var{extrafactor[1]}}{\\var{extrafactor[1]}}$ and the second fraction by $\\frac{\\var{extrafactor[0]}}{\\var{extrafactor[0]}}$ as follows.</p>\n<p><span data-jme-visible=\"addSub=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}+\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}+\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}+\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span></p>\n<p><span data-jme-visible=\"addSub=-1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}-\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}-\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}-\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span><span data-jme-visible=\"addSub=-1 and cf&lt;&gt;1\"></span></p>\n</div>\n<p></p>\n<div data-jme-visible=\"cf&lt;&gt;1 and extrafactor[0]=1\">\n<p>Since $\\var{den2}=\\var{cf}\\times \\var{extrafactor[1]}$ and the other denominator is $\\var{den1}$, the lowest common denominator will be the product $\\var{cf}\\times \\var{extrafactor[1]}$, that is, $\\var{ansden}$. Therefore, we will multiply the first fraction by $\\frac{\\var{extrafactor[1]}}{\\var{extrafactor[1]}}$ as follows.</p>\n<p><span data-jme-visible=\"addSub=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}+\\dfrac{\\var{num2}}{\\var{den2}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}+\\dfrac{\\var{num2}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}+\\var{num2}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span></p>\n<p><span data-jme-visible=\"addSub=-1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}-\\dfrac{\\var{num2}}{\\var{den2}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}-\\dfrac{\\var{num2}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}-\\var{num2}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span><span data-jme-visible=\"addSub=-1 and cf&lt;&gt;1\"></span></p>\n</div>\n<p></p>\n<div data-jme-visible=\"cf&lt;&gt;1 and extrafactor[1]=1\">\n<p>Since $\\var{den1}=\\var{cf}\\times\\var{extrafactor[0]}$ and the other denominator is $\\var{den2}$, the lowest common denominator will be the product $\\var{cf}\\times \\var{extrafactor[0]}$, that is, $\\var{ansden}$. Therefore, we will multiply the second fraction by $\\frac{\\var{extrafactor[0]}}{\\var{extrafactor[0]}}$ as follows.</p>\n<p><span data-jme-visible=\"addSub=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1}}{\\var{ansden}}+\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1}+\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span></p>\n<p><span data-jme-visible=\"addSub=-1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1}}{\\var{ansden}}-\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1}-\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span><span data-jme-visible=\"addSub=-1 and cf&lt;&gt;1\"></span></p>\n</div>\n<p></p>\n<p><span data-jme-visible=\"cf=1\">Since there are no common factors between the two denominators (other than $1$) the lowest common denominator will just be the product of the two denominators. Therefore, we will multiply the first fraction by $\\frac{\\var{extrafactor[1]}}{\\var{extrafactor[1]}}$ and the second fraction by $\\frac{\\var{extrafactor[0]}}{\\var{extrafactor[0]}}$ as follows.</span></p>\n<p><span data-jme-visible=\"addsub=1 and cf=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}+\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}+\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}+\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span></p>\n<p><span data-jme-visible=\"addsub=-1 and cf=1\">$\\begin{align*}\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}}{\\var{den2}}&amp;=\\dfrac{\\var{num1}\\times \\var{extrafactor[1]}}{\\var{den1}\\times \\var{extrafactor[1]}}-\\dfrac{\\var{num2}\\times \\var{extrafactor[0]}}{\\var{den2}\\times \\var{extrafactor[0]}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}}{\\var{ansden}}-\\dfrac{\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{num1*extrafactor[1]}-\\var{num2*extrafactor[0]}}{\\var{ansden}}\\\\[3pt]&amp;=\\dfrac{\\var{ansNum}}{\\var{ansden}}\\end{align*}$</span></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ansden": {"name": "ansden", "group": "Ungrouped variables", "definition": "cf*extrafactor[0]*extrafactor[1]", "description": "", "templateType": "anything", "can_override": false}, "num1": {"name": "num1", "group": "Ungrouped variables", "definition": "numbers[0]", "description": "", "templateType": "anything", "can_override": false}, "num2": {"name": "num2", "group": "Ungrouped variables", "definition": "numbers[1]", "description": "", "templateType": "anything", "can_override": false}, "addSub": {"name": "addSub", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "<p>adding or subtracting</p>", "templateType": "anything", "can_override": false}, "ansnum": {"name": "ansnum", "group": "Ungrouped variables", "definition": "if(addSub=1,num1*extrafactor[1]+num2*extrafactor[0],num1*extrafactor[1]-num2*extrafactor[0])", "description": "", "templateType": "anything", "can_override": false}, "numbers": {"name": "numbers", "group": "Ungrouped variables", "definition": "shuffle([2,3,5,7,11,13,17,19,23,29])[0..2]", "description": "", "templateType": "anything", "can_override": false}, "cf": {"name": "cf", "group": "Ungrouped variables", "definition": "random([1,1,1,2,4,8,3,9,5,25,7,11] except [num1,num2, num1^2, num2^2, num1^3, num2^3] )", "description": "<p>common factor of denominators</p>", "templateType": "anything", "can_override": false}, "extrafactor": {"name": "extrafactor", "group": "Ungrouped variables", "definition": "if(cf=1,shuffle([2,3,5,7] except [num1, num2])[0..2],shuffle([1,2,3,5,7] except [num1, num2])[0..2])", "description": "<p>extra factors in the denominators</p>", "templateType": "anything", "can_override": false}, "den1": {"name": "den1", "group": "Ungrouped variables", "definition": "cf*extrafactor[0]", "description": "", "templateType": "anything", "can_override": false}, "den2": {"name": "den2", "group": "Ungrouped variables", "definition": "cf*extrafactor[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["numbers", "num1", "num2", "addSub", "cf", "extrafactor", "den1", "den2", "ansden", "ansnum"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><span data-jme-visible=\"addSub=1\">$\\dfrac{\\var{num1}}{\\var{den1}}+\\dfrac{\\var{num2}}{\\var{den2}}$</span> <span data-jme-visible=\"addSub=-1\">$\\dfrac{\\var{num1}}{\\var{den1}}-\\dfrac{\\var{num2}}{\\var{den2}}$</span>&nbsp; $=$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansNum/ansden", "maxValue": "ansNum/ansden", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fractions/division and multiplication, different ways of presenting the same thing (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Students seem to not realise that $\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c&nbsp;\\div (b\\times a)\\ldots $ etc. This question is my attempt to help rectify this.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Without the use of a calculator and without actually calculating the values of each answer, which of the following are equal to $\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$?</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "primes[0]", "description": "", "templateType": "anything", "can_override": false}, "markss": {"name": "markss", "group": "Ungrouped variables", "definition": "[1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,1]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "primes[2]", "description": "", "templateType": "anything", "can_override": false}, "primes": {"name": "primes", "group": "Ungrouped variables", "definition": "shuffle([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131])", "description": "", "templateType": "anything", "can_override": false}, "choices": {"name": "choices", "group": "Ungrouped variables", "definition": "[\n  '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\var{c}$',\n    '$\\\\displaystyle\\\\frac{\\\\var{c}}{\\\\var{b}}\\\\times\\\\var{a}$',\n  '$\\\\displaystyle\\\\var{a}\\\\times\\\\frac{\\\\var{c}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{\\\\var{c}\\\\times\\\\var{a}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{\\\\var{a}\\\\times\\\\var{c}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\div\\\\var{b}$',\n    '$\\\\displaystyle(\\\\var{a}\\\\div\\\\var{b})\\\\times\\\\var{c}$',\n  '$\\\\displaystyle\\\\var{c}\\\\div\\\\var{b}\\\\times\\\\var{a}$',\n   '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\frac{\\\\var{c}}{1}$',\n   '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\times\\\\frac{1}{\\\\var{b}}$',\n  \n  '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{c}}\\\\times\\\\var{b}$',\n   '$\\\\displaystyle\\\\var{c}\\\\div(\\\\var{b}\\\\times\\\\var{a})$',\n    '$\\\\displaystyle\\\\frac{\\\\var{b}}{\\\\var{c}\\\\times\\\\var{a}}$',\n     '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\frac{1}{\\\\var{c}}$',\n   '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\div\\\\frac{1}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{1}{\\\\var{b}}\\\\times\\\\var{a}\\\\times\\\\var{c}$'\n]", "description": "<p>$\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c&nbsp;\\div (b\\times a)\\ldots $ etc</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "primes[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["primes", "a", "b", "c", "choices", "markss"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "stepsPenalty": "11", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Recall the following:</p>\n<ul>\n<li>the bar in a fraction is really just division. For example, $\\frac{1}{2}=1\\div 2$.</li>\n<li>the order of operations dictates that if a term contains multiplication and division, then you&nbsp;do the operations from left to right. For example, $4\\div 1&nbsp;\\times 4 = (4\\div 1 )\\times 4=16$.</li>\n<li>the order of multiplication is irrelevant. For example, $3\\times 4=4\\times 3$.</li>\n<li>a whole number can be written as a fraction over 1. For example, $3=\\frac{3}{1}$.</li>\n<li>division&nbsp;by a number is equivalent to multiplication by that numbers reciprocal. For example, $12\\div 3=12\\times \\frac{1}{3}$, and $3 \\div \\frac{2}{3}=3\\times \\frac{3}{2}$.</li>\n<li>multiplication by a number is equivalent to division by that numbers reciprocal. For example, $10\\times \\frac{1}{5}=10\\div 5$.</li>\n</ul>\n<p></p>\n<p>The above gives us (amoung other things)&nbsp;that&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$</td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{c}\\times\\var{a}\\div\\var{b}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\var{c}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle(\\var{a}\\div\\var{b})\\times\\var{c}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\frac{\\var{c}}{1}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}\\times\\var{c}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{c}\\times\\var{a}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{c}}{\\var{b}}\\times\\var{a}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{c}\\div\\var{b}\\times\\var{a}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\frac{\\var{c}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\var{c}\\div\\var{b}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\var{c}\\times\\frac{1}{\\var{b}}$</td>\n</tr>\n</tbody>\n</table>"}], "minMarks": 0, "maxMarks": "0", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "2", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": "{choices}", "matrix": "markss"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fractions: improper and mixed numerals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["Fractions", "fractions", "improper", "mixed numbers", "mixed numerals"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "b1+random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "(a3*c3+b3)/c3", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "ans4": {"name": "ans4", "group": "Ungrouped variables", "definition": "(a4*c4+b4)/c4", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "b4+random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "b3+random(1..7)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "b2+random(1..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "b1", "c1", "a2", "b2", "c2", "a3", "b3", "c3", "ans3", "a4", "b4", "c4", "ans4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Convert the following improper fractions to mixed numerals (also known as mixed numbers):</p>\n<p><em>Note: Write the whole number part in the first box&nbsp;and the fraction&nbsp;part&nbsp;in the second. <br/></em><em>For example, if your answer was $2\\frac{3}{4}$, enter $2$ in the first box and $3/4$ in the second.&nbsp;</em></p>\n<p>$\\displaystyle\\frac{\\var{a1*c1+b1}}{\\var{c1}}=$[[0]][[1]]</p>\n<p>$\\displaystyle\\frac{\\var{a2*c2+b2}}{\\var{c2}}=$[[2]][[3]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Do the division and write your remainder over the original denominator. Simplify the fraction if possible.</em>&nbsp;</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example,&nbsp;converting&nbsp;$\\frac{21}{9}$ into a mixed numeral, you ask yourself \"how many times does 9&nbsp;go into 21?\", it goes in twice&nbsp;(since $2\\times 9=18$ but $3\\times 9=27$), with a remainder of 3&nbsp;(since $21-18=3$). So we can write our answer as $2\\frac{3}{9}$ (which actually means $2+\\frac{3}{9}$). But notice we can simplify the fraction, so we should rewrite our answer as $2\\frac{1}{3}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Note: we could have cancelled common factors at the beginning.&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1", "maxValue": "a1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b1/c1", "maxValue": "b1/c1", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a2", "maxValue": "a2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b2/c2", "maxValue": "b2/c2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Convert the following mixed numerals to improper fractions:</p>\n<p>$\\var{a3}\\frac{\\var{b3}}{\\var{c3}}=$[[0]]</p>\n<p>$\\var{a4}\\frac{\\var{b4}}{\\var{c4}}=$[[1]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Multiply the whole number and the denominator, add the numerator, and put it all over the denominator.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example $2\\frac{3}{4}$ can be written as $\\frac{2\\times 4+3}{4}$ that is, $\\frac{11}{4}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>To understand why, realise that $2\\frac{3}{4}$ is shorthand for $2+\\frac{3}{4}$&nbsp;and&nbsp;if we want to add these numbers we need to&nbsp;have a common denominator (recall the denominator of a whole number is 1). Our working could look like this:</p>\n<p>\\[2\\tfrac{3}{4}=2+\\frac{3}{4}=\\frac{2\\times 4}{4}+\\frac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]</p>\n<p>but in practice we normally don't write anything more than \\[2\\tfrac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans3", "maxValue": "ans3", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans4", "maxValue": "ans4", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fractions: operations involving numerical fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["adding fractions", "dividing fractions", "multiplying fractions", "number fractions", "numerical fractions", "rational", "subtracting fractions"], "metadata": {"description": "<p>Add, subtract, multiply and divide numerical fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate the following and write your answer as a fraction or whole number (not a decimal). Use &nbsp;/ to signify a fraction or division, for example $\\frac{2}{3}$ is written 2/3.</p>", "advice": "<p>Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "numerical fractions", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "numerical fractions", "definition": "random(2..12 except [a,b])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "numerical fractions", "definition": "random(2..12 except a)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "numerical fractions", "definition": "random(2..12 except[a,b,c,10])", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "numerical fractions", "definition": "random(2..12 except f)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "numerical fractions", "definition": "random(2..12 except d)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "numerical fractions", "definition": "random(2..12 except [f,g])", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "numerical fractions", "definition": "random(2..12 except [f,g,h])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "numerical fractions", "variables": ["a", "b", "c", "d", "f", "g", "h", "j"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\frac{\\var{a}}{\\var{b}}+\\frac{\\var{c}}{\\var{b}}=$[[0]]</p>\n<p>$\\displaystyle\\frac{\\var{d}}{\\var{c}}-\\frac{\\var{a}}{\\var{c}}=$[[1]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Add the tops, leave the bottom the same.</em></p>\n<p></p>\n<p>These fractions have a common denominator (the number on the bottom). This means they are out of the same number of parts and can be compared easily, for example, it is clear $\\frac{2}{3}$ is less than $\\frac{5}{3}$ but not so clear that $\\frac{3}{5}$ is less than $\\frac{2}{3}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Let's say you need to evaluate $\\frac{2}{3}+\\frac{5}{3}$, in words this is 'two thirds plus five thirds', so how many thirds are there in total? Seven&nbsp;thirds!</p>\n<p style=\"text-align: left;\">So we have</p>\n<p style=\"text-align: center;\">\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]</p>\n<p>The same logic is used for subtraction. Suppose you had seven&nbsp;fourths and someone borrowed three fourths, then you are left with&nbsp;four&nbsp;fourths.&nbsp;</p>\n<p>That is</p>\n<p style=\"text-align: center;\">\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(a+c)/b", "maxValue": "(a+c)/b", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(d-a)/c", "maxValue": "(d-a)/c", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\simplify{{f}/{g}+{h}/{j}}=$[[0]]</p>\n<p>$\\displaystyle\\simplify{{h}/{f}-{j}/{g}}=$[[1]]</p>\n<p>$\\displaystyle \\frac{\\var{a}}{\\var{d}}+\\var{f}=$[[2]]</p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Rewrite the fractions so they have a common denominator. Then perform the addition or subtraction as required.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>If your question was $\\frac{5}{4}+\\frac{3}{8}$ we could rewrite the first fraction as $\\frac{10}{8}$ (by multiplying the top and bottom by 2) and then both fractions would have a denominator of 8. At this point, we can perform the addition. Our working might&nbsp;look like this:</p>\n<p style=\"text-align: center;\">\\[\\frac{5}{4}+\\frac{3}{8}=\\frac{5\\times 2}{4\\times 2}+\\frac{3}{8}=\\frac{10}{8}+\\frac{3}{8}=\\frac{13}{8}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Often we need to rewrite both fractions to get a common denominator, for instance, $\\frac{5}{4}-\\frac{2}{3}$. We could multiply the first fraction by 3 on the top and bottom, so that it's denominator was 12, and then multiply the second fraction by 4 on the top and bottom&nbsp;so that it also had a denominator of 12. Then we could perform the subtraction. Our working might look like this:</p>\n<p style=\"text-align: center;\">\\[\\frac{5}{4}-\\frac{2}{3}=\\frac{5\\times 3}{4\\times 3}-\\frac{2\\times 4}{3\\times 4}=\\frac{15}{12}-\\frac{8}{12}=\\frac{7}{12}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Also, recall that whole numbers are just fractions with a denominator of 1, for example $3=\\frac{3}{1}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In general, the best denominator is the lowest common multiple (LCM) of the two denominators.&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(f*j+h*g)/(g*j)", "maxValue": "(f*j+h*g)/(g*j)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(h*g-j*f)/(f*g)", "maxValue": "(h*g-j*f)/(f*g)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(a+f*d)/(d)", "maxValue": "(a+f*d)/(d)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times \\frac{\\var{c}}{\\var{d}}=$[[0]]</p>\n<p>$\\displaystyle -\\frac{\\var{f}}{\\var{j}}\\times \\var{d}=$[[1]]</p>\n<p></p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Multiply the tops and the bottoms.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example&nbsp;</p>\n<p style=\"text-align: center;\">\\[\\frac{4}{5}\\times \\frac{2}{3}=\\frac{4\\times 2}{5 \\times 3}=\\frac{8}{15}\\]</p>\n<p style=\"text-align: center;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Also recall that whole numbers are just fractions with a denominator of 1, for example $7=\\frac{7}{1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(a*c)/(b*d)", "maxValue": "(a*c)/(b*d)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-f*d/j", "maxValue": "-f*d/j", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle&nbsp;\\frac{\\var{f}}{\\var{h}}\\div \\frac{\\var{g}}{\\var{j}}=$[[0]]</p>\n<p>$\\displaystyle \\frac{\\var{b}}{\\var{c}}\\div \\var{d}=$[[1]]</p>\n<p>$\\displaystyle \\var{j}\\div \\left(\\frac{\\var{-d}}{\\var{f}}\\right)=$[[2]]</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Flip the second fraction and then multiply.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>Flipping a fraction is also known as taking the reciprocal of the fraction (or inverting a fraction). Note that a whole number is also a fraction with a denominator of 1, for example, $6=\\frac{6}{1}$.</p>\n<p>How do you find half of a number? You could 'divide it by 2', or you could 'multiply by $\\frac{1}{2}$. Notice that $\\frac{1}{2}$ is the reciprocal of 2. When we divide by a number this is actually the same as multiplying by its reciprocal.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Suppose you need to evaluate $\\frac{3}{7}\\div\\frac{5}{4}$. Recall this is the same as asking 'how many $\\frac{5}{4}$s are in $\\frac{3}{7}$?', but that doesn't seem to be very helpful here! What is helpful is realising that dividing by $\\frac{5}{4}$ is the same as multiplying by $\\frac{4}{5}$. Our working could&nbsp;look like this</p>\n<p>\\[\\frac{3}{7}\\div\\frac{5}{4}=\\frac{3}{7}\\times\\frac{4}{5}=\\frac{3\\times 4}{7\\times 5}=\\frac{12}{35}\\]</p>\n<p>&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(f*j)/(h*g)", "maxValue": "(f*j)/(h*g)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b/(c*d)", "maxValue": "b/(c*d)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "j*f/(-d)", "maxValue": "j*f/(-d)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle \\frac{\\frac{\\var{b}}{\\var{c}}}{ \\frac{\\var{a}}{\\var{d}}}=$[[0]]</p>\n<p></p>\n<p>$\\displaystyle \\frac{\\frac{\\var{d}}{\\var{g}}}{\\var{f}}=$[[1]]</p>\n<p></p>\n<p>$\\displaystyle \\frac{\\var{j}}{\\frac{\\var{h}}{\\var{c}}}=$[[2]]</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>The fraction bar means division.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>The fraction $\\frac{2}{3}$ means 2 divided by 3. So these questions are just division questions! It is important to note which fraction bar is big and which are small, so you know the order of the divisions. &nbsp;</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Here are some examples:</p>\n<p>\\[\\frac{7}{\\frac{5}{6}}=7\\div\\frac{5}{6} =7\\times\\frac{6}{5}=\\frac{42}{5}\\]</p>\n<p>\\[\\frac{\\frac{7}{5}}{6}=\\frac{7}{5}\\div 6=\\frac{7}{5}\\times \\frac{1}{6}=\\frac{7}{30}\\]</p>\n<p>\\[\\frac{\\frac{9}{11}}{\\frac{5}{3}}=\\frac{9}{11}\\div\\frac{5}{3}=\\frac{9}{11}\\times \\frac{3}{5}=\\frac{27}{55}\\]</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(b*d)/(a*c)", "maxValue": "(b*d)/(a*c)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "d/(g*f)", "maxValue": "d/(g*f)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "j*c/h", "maxValue": "j*c/h", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Fraction of a pizza remaining (4)", "extensions": ["written-number"], "custom_part_types": [], "resources": [["question-resources/pizza1.png", "/srv/numbas/media/question-resources/pizza1.png"], ["question-resources/pizza2.png", "/srv/numbas/media/question-resources/pizza2.png"], ["question-resources/pizza3.png", "/srv/numbas/media/question-resources/pizza3.png"], ["question-resources/pizza4.png", "/srv/numbas/media/question-resources/pizza4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Adelle Colbourn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2083/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["LANTITE"], "metadata": {"description": "<p>NC NA Non-calculator and Number and Algebra strand. This number assesses students' ability to add and subtract very simple fractions. Students need two add two fractions (possibly mixed numbers) involving quarters and/or halves, then subtract their answer from a whole number. The answer must be entered in simplest form and will always be less than one.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p><strong>Do not use a calculator for this question.</strong></p>\n<p>A parent bought their two children {written_number(number)} mini-pizzas at the school fair. The older child {frac1} and the younger child {frac2}.</p>", "advice": "<p>{advice}</p>\n<p>The diagrams below show the whole mini-pizzas, the amount eaten and the amount left.</p>\n<p>{image('resources/question-resources/'+image,30)}</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"scenarios": {"name": "scenarios", "group": "Ungrouped variables", "definition": "json_decode(safe(\"[\\n{\\n\\\"num\\\":2,\\n\\\"f1\\\":\\\"ate three quarters of a mini-pizza\\\",\\n\\\"f2\\\":\\\"ate half a mini-pizza\\\",\\n\\\"ans\\\":\\\"3/4\\\",\\n\\\"img\\\":\\\"pizza4.png\\\",\\n\\\"adv\\\":\\\"First we need to add $\\\\\\\\frac{3}{4}$ and $\\\\\\\\frac{1}{2}$.<br>We know that $\\\\\\\\frac{1}{2}+\\\\\\\\frac{1}{2}=1$ so $\\\\\\\\frac{3}{4}+\\\\\\\\frac{1}{2}$ will be $\\\\\\\\frac{1}{4}$ more than that.<br>$\\\\\\\\frac{3}{4}+\\\\\\\\frac{1}{2}=1\\\\\\\\frac{1}{4}$.<br>So the two children together ate $1\\\\\\\\frac{1}{4}$ mini-pizzas out of the two mini-pizzas that they started with.<br>This means that $\\\\\\\\frac{3}{4}$ of a mini-pizza remained.\\\"\\n},\\n{\\n\\\"num\\\":2,\\n\\\"f1\\\":\\\"ate three quarters of a mini-pizza\\\",\\n\\\"f2\\\":\\\"also ate three quarters of a mini-pizza\\\",\\n\\\"ans\\\":\\\"1/2\\\",\\n\\\"img\\\":\\\"pizza2.png\\\",\\n\\\"adv\\\":\\\"First we need to add $\\\\\\\\frac{3}{4}$ and $\\\\\\\\frac{3}{4}$.<br>We know that $\\\\\\\\frac{1}{2}+\\\\\\\\frac{1}{2}=1$ so $\\\\\\\\frac{3}{4}+\\\\\\\\frac{3}{4}$ will be $\\\\\\\\frac{1}{2}$ more than that (adding the extra quarters).<br>$\\\\\\\\frac{3}{4}+\\\\\\\\frac{3}{4}=1\\\\\\\\frac{1}{2}$.<br>So the two children together ate $1\\\\\\\\frac{1}{2}$ mini-pizzas out of the two mini-pizzas that they started with.<br>This means that $\\\\\\\\frac{1}{2}$ of a mini-pizza remained.\\\"\\n},\\n{\\n\\\"num\\\":3,\\n\\\"f1\\\":\\\"ate one and a half mini-pizzas\\\",\\n\\\"f2\\\":\\\"ate one and a quarter mini-pizzas\\\",\\n\\\"ans\\\":\\\"1/4\\\",\\n\\\"img\\\":\\\"pizza3.png\\\",\\n\\\"adv\\\":\\\"First we need to add $1\\\\\\\\frac{1}{2}$ and $1\\\\\\\\frac{1}{4}$.<br>We know that $\\\\\\\\frac{1}{2}+\\\\\\\\frac{1}{4}=\\\\\\\\frac{3}{4}$ so $1\\\\\\\\frac{1}{2}+1\\\\\\\\frac{1}{4}$ will be $2\\\\\\\\frac{3}{4}$.<br>So the two children together ate $2\\\\\\\\frac{3}{4}$ mini-pizzas out of the three mini-pizzas that they started with.<br>This means that $\\\\\\\\frac{1}{4}$ of a mini-pizza remained.\\\"\\n},\\n{\\n\\\"num\\\":3,\\n\\\"f1\\\":\\\"ate one and three quarters of the mini-pizzas\\\",\\n\\\"f2\\\":\\\"ate three quarters of a mini-pizza\\\",\\n\\\"ans\\\":\\\"1/2\\\",\\n\\\"img\\\":\\\"pizza1.png\\\",\\n\\\"adv\\\":\\\"First we need to add $1\\\\\\\\frac{3}{4}$ and $\\\\\\\\frac{3}{4}$.<br>We know that $\\\\\\\\frac{1}{2}+\\\\\\\\frac{1}{2}=1$ so $\\\\\\\\frac{3}{4}+\\\\\\\\frac{3}{4}$ will be $\\\\\\\\frac{1}{2}$ more than that (adding the extra quarters).<br>$\\\\\\\\frac{3}{4}+\\\\\\\\frac{3}{4}=1\\\\\\\\frac{1}{2}$.<br>Therefore $1\\\\\\\\frac{3}{4}+\\\\\\\\frac{3}{4}$ will be $2\\\\\\\\frac{1}{2}$.<br>So the two children together ate $2\\\\\\\\frac{1}{2}$ mini-pizzas out of the three mini-pizzas that they started with.<br>This means that $\\\\\\\\frac{1}{2}$ of a mini-pizza remained.\\\"\\n}\\n]\"))", "description": "<p>The list of all pizza scenarios, each with the number of pizzas, fraction each child ate, answer and advice (worked solution).</p>", "templateType": "json", "can_override": false}, "scenario": {"name": "scenario", "group": "Ungrouped variables", "definition": "random(scenarios)", "description": "<p>The randomly selected pizza scenario, with the number of pizzas, fraction each child ate, answer and advice (worked solution).</p>", "templateType": "anything", "can_override": false}, "number": {"name": "number", "group": "Ungrouped variables", "definition": "scenario[\"num\"]", "description": "<p>The number of pizzas that the parent bought.</p>", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "scenario[\"ans\"]", "description": "<p>The correct answer - the fraction of pizza remaining.</p>", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "scenario[\"adv\"]", "description": "<p>The advice (worked solution) for the randomly selected question.</p>", "templateType": "anything", "can_override": false}, "frac1": {"name": "frac1", "group": "Ungrouped variables", "definition": "scenario[\"f1\"]", "description": "<p>The fraction of a pizza (or pizzas) that the older child ate.</p>", "templateType": "anything", "can_override": false}, "frac2": {"name": "frac2", "group": "Ungrouped variables", "definition": "scenario[\"f2\"]", "description": "<p>The fraction of a pizza (or pizzas) that the younger child ate.</p>", "templateType": "anything", "can_override": false}, "image": {"name": "image", "group": "Ungrouped variables", "definition": "scenario[\"img\"]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["scenarios", "scenario", "number", "frac1", "frac2", "answer", "advice", "image"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What fraction of a pizza remained altogether? Enter your answer in simplest form using the / key.</p>", "answer": "{answer}", "displayAnswer": "{answer}", "matchMode": "regex"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Converting between Mixed Numbers and Improper Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "rulesets": {}, "variable_groups": [], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "gcd", "ab", "cd", "f", "g", "h", "gcd_gh", "g_coprime", "h_coprime", "num", "l", "j", "k", "gcd_jk", "j_coprime", "k_coprime", "num2", "s", "t", "rounds", "gap1", "gcd_gap1t", "gap1_coprime", "t_coprime", "x", "y", "roundx", "gap4", "gcda", "gcdb", "gcd_gap4y", "gap4_coprime", "y_coprime"], "metadata": {"description": "<p>This question tests the student's ability to identify equivalent fractions&nbsp;through spotting a fraction which is not equivalent amongst a list of otherwise equivalent fractions. It also tests the students ability to convert mixed numbers into their equivalent improper fractions. It then does the reverse and tests their ability to convert an improper fraction into an equivalent mixed number.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "<h4><strong></strong><strong>c)</strong></h4>\n<p>A <em>mixed number</em> is a number consisting of an integer and a proper fraction, i.e. a number in the form $&nbsp;a \\displaystyle \\frac{b}{c}$ where $a$ is an integer and $\\displaystyle\\frac{b}{c}$ is a proper fraction: $b$ is smaller than $c$.</p>\n<p>An <em>improper fraction</em> is a fraction where the numerator is larger than the denominator, i.e. a number of the form $\\displaystyle\\frac{d}{e}$ where&nbsp;the numerator, $d$, is greater than the denominator, $e$.&nbsp;</p>\n<p>To convert a mixed number into an improper fraction, multiply the integer part of the mixed number, $a$, by the denominator, $c$.</p>\n<p>The numerator of the improper fraction will be equal to this&nbsp;added to what was already on the numerator of the proper fraction.</p>\n<p>The denominator of the proper fraction will stay the same when it converts to an improper fraction to give a final answer of</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{({a}\\times{c})+b}{c}$.</p>\n<p><em>i)</em></p>\n<p>\\[<br/>{\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}} = \\frac{({\\var{f}}\\times{\\var{h_coprime}})+{\\var{g_coprime}}}{{\\var{h_coprime}}}=\\simplify{{num}/{h_coprime}}\\text{.}<br/>\\]&nbsp;</p>\n<p><em>ii)</em></p>\n<p>\\[<br/>{\\var{l}\\frac{\\var{j_coprime}}{\\var{k_coprime}}} = \\frac{({\\var{l}}\\times{\\var{k_coprime}})+{\\var{j_coprime}}}{{\\var{k_coprime}}} =\\simplify{{num2}/{k_coprime}}\\text{.}<br/>\\]</p>\n<p></p>\n<h4><strong>d)</strong></h4>\n<p>To convert an improper fraction into a mixed number, you have to think about how many times the denominator can go into the numerator, and whatever is left over becomes the new numerator of the proper fraction.&nbsp;</p>\n<p><em>i)</em>&nbsp;$\\displaystyle\\frac{\\var{s}}{\\var{t}}$: &nbsp;</p>\n<p>$\\var{s}$ can be divided by $\\var{t}$ fully $\\var{rounds}$ {if(rounds=1,\"time\",\"times\")}, with $\\var{gap1}$ remaining.</p>\n<p>$\\var{rounds}$ becomes the integer part of the mixed number and $\\var{gap1}$ becomes the new numerator of the proper fraction.&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{s}}{\\var{t}} = \\var{rounds}\\frac{\\var{gap1}}{\\var{t}}$.</p>\n<p style=\"text-align: left;\">It may be possible to simplify the proper fraction further by finding the highest common divisor of the numerator and denominator. In this question, this is $\\var{gcd_gap1t}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_gap1t = 1\">Therefore, it is not possible to simplify this further, and the final answer shall be</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_gap1t &gt; 1\">Simplifying by this value gives the final answer</span></p>\n<p style=\"text-align: center;\">$\\displaystyle\\var{rounds}\\frac{\\var{gap1_coprime}}{\\var{t_coprime}}$.</p>\n<p><em>ii)</em>&nbsp;$\\displaystyle\\frac{\\var{x}}{\\var{y}}$:</p>\n<p>$\\var{x}$ can be divided by $\\var{y}$ fully $\\var{roundx}$ {if(roundx=1,\"time\",\"times\")}, with $\\var{gap4}$ left over.</p>\n<p>$\\var{roundx}$ becomes the integer part of the mixed number and $\\var{gap4}$ becomes the new numerator of the proper fraction.&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{x}}{\\var{y}} = \\var{roundx}\\frac{\\var{gap4}}{\\var{y}}$.</p>\n<p style=\"text-align: left;\">It may be possible to simplify the proper fraction further by finding the highest common divisor of the numerator and denominator. In this question, this is $\\var{gcd_gap4y}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_gap4y = 1\">Therefore, it is not possible to simplify this further, and the final answer shall be</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_gap4y &gt; 1\">Simplifying by this value gives the final answer</span></p>\n<p style=\"text-align: center;\">$\\displaystyle\\var{roundx}\\frac{\\var{gap4_coprime}}{\\var{y_coprime}}$.</p>", "statement": "", "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n", "js": ""}, "variables": {"x": {"templateType": "anything", "description": "<p>Random number between 15 and 35</p>", "name": "x", "group": "Ungrouped variables", "definition": "random(11..29 except 12 except 14 except 16 except 18 except 20 except 24 except 25 except 27 except 28 except 21 except 15)"}, "k_coprime": {"templateType": "anything", "description": "<p>PART C</p>", "name": "k_coprime", "group": "Ungrouped variables", "definition": "k/gcd_jk"}, "g": {"templateType": "anything", "description": "<p>Random number between 1 and 15</p>", "name": "g", "group": "Ungrouped variables", "definition": "random(1..15 except h)"}, "g_coprime": {"templateType": "anything", "description": "<p>PART C</p>", "name": "g_coprime", "group": "Ungrouped variables", "definition": "g/gcd_gh"}, "gcd": {"templateType": "anything", "description": "<p>greatest common divisor of ab and cd</p>", "name": "gcd", "group": "Ungrouped variables", "definition": "gcd(ab,cd)"}, "d": {"templateType": "randrange", "description": "<p>Random number from 1 to 12</p>", "name": "d", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "ab": {"templateType": "anything", "description": "<p>variable a times variable b.</p>", "name": "ab", "group": "Ungrouped variables", "definition": "a*b"}, "gcd_gh": {"templateType": "anything", "description": "<p>PART C</p>", "name": "gcd_gh", "group": "Ungrouped variables", "definition": "gcd(g,h)"}, "a": {"templateType": "randrange", "description": "<p>A random number from 1-12.</p>", "name": "a", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "j_coprime": {"templateType": "anything", "description": "<p>PART C</p>", "name": "j_coprime", "group": "Ungrouped variables", "definition": "j/gcd_jk"}, "gcd_gap4y": {"templateType": "anything", "description": "<p>PART D</p>", "name": "gcd_gap4y", "group": "Ungrouped variables", "definition": "gcd(gap4,y)"}, "gcd_gap1t": {"templateType": "anything", "description": "", "name": "gcd_gap1t", "group": "Ungrouped variables", "definition": "gcd(gap1,t)"}, "num": {"templateType": "anything", "description": "<p>numerator for the improper fraction c(i)</p>", "name": "num", "group": "Ungrouped variables", "definition": "((h_coprime*f)+g_coprime)"}, "s": {"templateType": "anything", "description": "<p>Random number between 15 and 30</p>", "name": "s", "group": "Ungrouped variables", "definition": "random(11..29 except 12 except 14 except 16 except 18 except 20 except 24 except 25 except 27 except 28 except 21 except 15)"}, "h": {"templateType": "randrange", "description": "<p>Random number between 1 and 24</p>", "name": "h", "group": "Ungrouped variables", "definition": "random(10..24#1)"}, "gcda": {"templateType": "anything", "description": "", "name": "gcda", "group": "Ungrouped variables", "definition": "gcd({num2},{k_coprime})"}, "t_coprime": {"templateType": "anything", "description": "", "name": "t_coprime", "group": "Ungrouped variables", "definition": "t/gcd_gap1t"}, "h_coprime": {"templateType": "anything", "description": "<p>PART C</p>", "name": "h_coprime", "group": "Ungrouped variables", "definition": "h/gcd_gh"}, "b": {"templateType": "randrange", "description": "<p>A random number from 1 to 12</p>", "name": "b", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "gap4_coprime": {"templateType": "anything", "description": "", "name": "gap4_coprime", "group": "Ungrouped variables", "definition": "gap4/gcd_gap4y"}, "y_coprime": {"templateType": "anything", "description": "", "name": "y_coprime", "group": "Ungrouped variables", "definition": "y/gcd_gap4y"}, "rounds": {"templateType": "anything", "description": "<p>Rounded down version of (variable s)/(variable t)</p>", "name": "rounds", "group": "Ungrouped variables", "definition": "floor(s/t)"}, "t": {"templateType": "randrange", "description": "<p>Random number between 5 and 15</p>", "name": "t", "group": "Ungrouped variables", "definition": "random(5..10#1)"}, "gap1": {"templateType": "anything", "description": "<p>numerator for d(i)</p>", "name": "gap1", "group": "Ungrouped variables", "definition": "(s-(t*rounds))"}, "y": {"templateType": "randrange", "description": "<p>Random number between 2 and 20</p>", "name": "y", "group": "Ungrouped variables", "definition": "random(5..10#1)"}, "l": {"templateType": "randrange", "description": "<p>Random number from 1 to 12</p>", "name": "l", "group": "Ungrouped variables", "definition": "random(2..6#1)"}, "gap4": {"templateType": "anything", "description": "<p>Solution for gap 4 on part d(ii)</p>", "name": "gap4", "group": "Ungrouped variables", "definition": "(x-(y*roundx))"}, "j": {"templateType": "anything", "description": "<p>Random number from 1 to 12</p>", "name": "j", "group": "Ungrouped variables", "definition": "random(1..12 except k)"}, "cd": {"templateType": "anything", "description": "<p>variable c times variable d</p>", "name": "cd", "group": "Ungrouped variables", "definition": "c*d"}, "gcd_jk": {"templateType": "anything", "description": "<p>PART C</p>", "name": "gcd_jk", "group": "Ungrouped variables", "definition": "gcd(j,k)"}, "c": {"templateType": "randrange", "description": "<p>Random number from 1 to 12</p>", "name": "c", "group": "Ungrouped variables", "definition": "random(1..12#1)"}, "roundx": {"templateType": "anything", "description": "<p>Rounded down solution to x/y</p>", "name": "roundx", "group": "Ungrouped variables", "definition": "floor(x/y)"}, "gap1_coprime": {"templateType": "anything", "description": "", "name": "gap1_coprime", "group": "Ungrouped variables", "definition": "gap1/gcd_gap1t"}, "k": {"templateType": "randrange", "description": "<p>Random number from 1 to 12</p>", "name": "k", "group": "Ungrouped variables", "definition": "random(10..24#1)"}, "gcdb": {"templateType": "anything", "description": "<p>gcd of num and h</p>", "name": "gcdb", "group": "Ungrouped variables", "definition": "gcd({num},{h_coprime})"}, "f": {"templateType": "randrange", "description": "<p>Random number between 1 and 5</p>", "name": "f", "group": "Ungrouped variables", "definition": "random(2..5#1)"}, "num2": {"templateType": "anything", "description": "<p>numerator of c(ii)</p>", "name": "num2", "group": "Ungrouped variables", "definition": "((l*k_coprime)+j_coprime)"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "<p>Write these mixed numbers as their improper fraction equivalents and reduce them down to their simplest form.&nbsp;</p>\n<p><em>i)</em><strong>&nbsp;</strong></p>\n<p>$\\displaystyle{\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}} =$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction>.</p>\n<p><em>ii)</em></p>\n<p>$\\displaystyle{\\var{l}\\frac{\\var{j_coprime}}{\\var{k_coprime}}} =$&nbsp;<fraction> <numerator>[[2]]</numerator> <denominator>[[3]]</denominator> </fraction>.</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "num/gcdb", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "num/gcdb", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "h_coprime/gcdb", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "h_coprime/gcdb", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "num2/gcda", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "num2/gcda", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "k_coprime/gcda", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "k_coprime/gcda", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "<p>Write these improper fractions as their mixed number equivalent.</p>\n<p><em>i)</em>&nbsp;</p>\n<p>$\\displaystyle\\frac{\\var{s}}{\\var{t}} =$&nbsp;[[0]]&nbsp;<fraction> <numerator>[[1]]</numerator> <denominator>[[2]]</denominator> </fraction>.</p>\n<p><em>ii)</em></p>\n<p>$\\displaystyle\\frac{\\var{x}}{\\var{y}} =$&nbsp;[[3]]&nbsp;<fraction> <numerator>[[4]]</numerator> <denominator>[[5]]</denominator> </fraction>.</p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "rounds", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "rounds", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "gap1_coprime", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "gap1_coprime", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "t_coprime", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "t_coprime", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "roundx", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "roundx", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "gap4_coprime", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "gap4_coprime", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "marks": 1, "correctAnswerStyle": "plain", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "y_coprime", "allowFractions": false, "mustBeReducedPC": 0, "minValue": "y_coprime", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], 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"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "b1+random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "(a3*c3+b3)/c3", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, 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"b3+random(1..7)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "b2+random(1..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "b1", "c1", "a2", "b2", "c2", "a3", "b3", "c3", "ans3", "a4", "b4", "c4", "ans4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Convert the following improper fractions to mixed numerals (also known as mixed numbers):</p>\n<p><em>Note: Write the whole number part in the first box&nbsp;and the fraction&nbsp;part&nbsp;in the second. <br/></em><em>For example, if your answer was $2\\frac{3}{4}$, enter $2$ in the first box and $3/4$ in the second.&nbsp;</em></p>\n<p>$\\displaystyle\\frac{\\var{a1*c1+b1}}{\\var{c1}}=$[[0]][[1]]</p>\n<p>$\\displaystyle\\frac{\\var{a2*c2+b2}}{\\var{c2}}=$[[2]][[3]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Do the division and write your remainder over the original denominator. Simplify the fraction if possible.</em>&nbsp;</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example,&nbsp;converting&nbsp;$\\frac{21}{9}$ into a mixed numeral, you ask yourself \"how many times does 9&nbsp;go into 21?\", it goes in twice&nbsp;(since $2\\times 9=18$ but $3\\times 9=27$), with a remainder of 3&nbsp;(since $21-18=3$). So we can write our answer as $2\\frac{3}{9}$ (which actually means $2+\\frac{3}{9}$). But notice we can simplify the fraction, so we should rewrite our answer as $2\\frac{1}{3}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Note: we could have cancelled common factors at the beginning.&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1", "maxValue": "a1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b1/c1", "maxValue": "b1/c1", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a2", "maxValue": "a2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b2/c2", "maxValue": "b2/c2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Convert the following mixed numerals to improper fractions:</p>\n<p>$\\var{a3}\\frac{\\var{b3}}{\\var{c3}}=$[[0]]</p>\n<p>$\\var{a4}\\frac{\\var{b4}}{\\var{c4}}=$[[1]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Multiply the whole number and the denominator, add the numerator, and put it all over the denominator.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example $2\\frac{3}{4}$ can be written as $\\frac{2\\times 4+3}{4}$ that is, $\\frac{11}{4}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>To understand why, realise that $2\\frac{3}{4}$ is shorthand for $2+\\frac{3}{4}$&nbsp;and&nbsp;if we want to add these numbers we need to&nbsp;have a common denominator (recall the denominator of a whole number is 1). Our working could look like this:</p>\n<p>\\[2\\tfrac{3}{4}=2+\\frac{3}{4}=\\frac{2\\times 4}{4}+\\frac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]</p>\n<p>but in practice we normally don't write anything more than \\[2\\tfrac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans3", "maxValue": "ans3", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans4", "maxValue": "ans4", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Work with Mixed Numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}, {"name": "Alissa Gerber", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9861/"}], "tags": [], "metadata": {"description": "<p>Add, subtract, multiply of divide two mixed numbers. This question has no advice at this stage.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate \\[\\var{whole1} \\var[fractionNumbers]{frac1} \\var{(latexop)} \\var{whole2} \\var[fractionNumbers]{frac2} \\]</p>\n<p>Give your answer as a mixed number in simplest form.</p>\n<p>Note: to enter a fraction, say \\(\\dfrac23\\), type <code>2/3</code></p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"whole1": {"name": "whole1", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "whole2": {"name": "whole2", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(n2..10)", "description": "", "templateType": "anything", "can_override": false}, "frac1": {"name": "frac1", "group": "Ungrouped variables", "definition": "n1/d1", "description": "", "templateType": "anything", "can_override": false}, "frac2": {"name": "frac2", "group": "Ungrouped variables", "definition": "n2/d2", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(n1..10)", "description": "", "templateType": "anything", "can_override": false}, "opidx": {"name": "opidx", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "<p>0 = +,&nbsp;1 = -,&nbsp;2 = *,&nbsp;3 = /</p>", "templateType": "anything", "can_override": false}, "latexop": {"name": "latexop", "group": "Ungrouped variables", "definition": "latex(['+','-','\\\\times','\\\\div'][opidx])", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "eval(expression(\n  \"(\" + whole1 + \"+\" + frac1 + \")\" +\n  [\"+\",\"-\",\"*\",\"/\"][opidx] +\n  \"(\"+ whole2 + \"+\" + frac2 + \")\"\n))", "description": "", "templateType": "anything", "can_override": false}, "answhole": {"name": "answhole", "group": "Ungrouped variables", "definition": "floor(ans)", "description": "", "templateType": "anything", "can_override": false}, "ansfrac": {"name": "ansfrac", "group": "Ungrouped variables", "definition": "ans-answhole", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["whole1", "whole2", "n1", "n2", "d1", "d2", "frac1", "frac2", "opidx", "latexop", "ans", "answhole", "ansfrac"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write the answer as a mixed numeral, with the whole number in the first box, and the fractional part in the second box.</p>\n<p>Answer = [[0]][[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "whole", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{answhole}", "maxValue": "{answhole}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "fraction", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ansfrac}", "maxValue": "{ansfrac}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": "50", "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Division of fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "metadata": {"description": "<p>Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": [], "advice": "<h4 style=\"text-align: left;\">a)</h4>\n<p>When faced with dividing fractions, it much easier to switch one of the fractions around and multiply them together instead of divide them.</p>\n<p style=\"text-align: center;\">\\[ \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}} \\right) \\equiv \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\times\\frac{\\var{j_coprime}}{\\var{h_coprime}} \\right) = \\frac{\\var{fj}}{\\var{gh}} \\]</p>\n<p>Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd1}$.&nbsp;</p>\n<p>This gives a final answer of $\\displaystyle\\simplify{{fj}/{gh}}$.</p>\n<p></p>\n<h4 style=\"text-align: left;\">b)</h4>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}} \\equiv \\left( \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\times\\frac{\\var{j1_coprime}}{\\var{h1_coprime}} \\right)=\\frac{\\var{f1j1}}{\\var{g1h1}} \\]</p>\n<p>Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd2}$.</p>\n<p>This gives a final answer of $\\displaystyle\\simplify{{f1j1}/{g1h1}}$.</p>\n<p></p>\n<h4>c)</h4>\n<p style=\"text-align: center;\">\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}} \\]</p>\n<p>The first thing to do is to change the mixed numbers into improper fractions.</p>\n<p>An improper fraction is a fraction where the numerator is greater than the denominator. To change a mixed fraction to an improper fraction, multiply the integer part of the mixed number by the denominator, and add it to the existing numerator to make the new numerator of the improper fraction. The denominator will stay the same.&nbsp;</p>\n<p style=\"text-align: center;\">\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\equiv\\frac{(\\var{f3}\\times\\var{h3_coprime})+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3}+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}} \\]</p>\n<p style=\"text-align: center;\">\\[ {\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}\\equiv\\frac{(\\var{f4}\\times\\var{h4_coprime})+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4}+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]</p>\n<p>We now have our mixed numbers as improper fractions.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]</p>\n<p>Now, use the same method as in parts <strong>a)</strong> and <strong>b)</strong> to divide by switching around one fraction and changing the division symbol to multiplication.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}}\\equiv\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\times\\frac{\\var{h4_coprime}}{\\var{f4h4+g4_coprime}}=\\frac{\\var{num}}{\\var{denom}} \\]</p>\n<p>Finally, the last thing to do is to simplify your answer down by finding the highest common divisor in the numerator and denominator, which in this case is $\\var{gcd3}$.</p>\n<p>By doing this, you will get a final answer of</p>\n<p style=\"text-align: center;\">\\[ \\simplify{{num}/{denom}} \\]</p>\n<h4>d)</h4>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} \\]</p>\n<p style=\"text-align: left;\">Consider the denominator first, as following the rules of BODMAS, you should address brackets first.</p>\n<p style=\"text-align: left;\">You need to get a common denominator for both terms on the denominator, like this:</p>\n<p style=\"text-align: center;\">\\[ \\var{b}\\times\\frac{\\var{d}}{\\var{d}} = \\frac{\\var{bd}}{\\var{d}} \\]</p>\n<p style=\"text-align: left;\">This now allows you to complete the addition or subtraction as both terms have a common denominator.&nbsp;</p>\n<p style=\"text-align: center;\">\\[ {\\simplify[all,!collectNumbers]{{bd}/{d}-{c}/{d}}} = \\frac{\\var{bd_c}}{\\var{d}} \\]</p>\n<p style=\"text-align: left;\">This means that the expression is now:</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}} \\]</p>\n<p style=\"text-align: left;\">Dealing with this requires a bit of manipulation. You have to change&nbsp;the divisor of the denominator to be a mulitplier of the numerator. The denominator, ${\\var{bd_c}}$ was being divided by ${\\var{d}}$ but by flipping it around, the numerator, ${\\var{a}}$ will be mulitplied by&nbsp;${\\var{d}}$. The value of the expression remains the same.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}}\\equiv \\frac{(\\var{a})\\times(\\var{d})}{\\var{bd_c}}= \\frac{\\var{ad}}{\\var{bd_c}} \\]</p>\n<p style=\"text-align: left;\">From this, you can try to cancel the expression down by finding the highest common factor of the numerator and denominator, to give a final answer of</p>\n<p style=\"text-align: center;\">\\[ \\simplify{{ad}/{bd_c}} \\]</p>", "variable_groups": [{"name": "part d", "variables": ["a", "b", "c", "d", "bd", "ad", "gcd", "ad_gcd", "bcd_gcd", "bd_c"]}, {"name": "part a", "variables": ["f", "g", "f_coprime", "g_coprime", "h", "j", "h_coprime", "j_coprime", "fj", "gh", "gcd1"]}, {"name": "part b", "variables": ["f1", "g1", "f1_coprime", "g1_coprime", "h1", "j1", "h1_coprime", "j1_coprime", "f1j1", "g1h1", "gcd2"]}, {"name": "part c", "variables": ["f3", "g3", "h3", "g3_coprime", "h3_coprime", "f4", "g4", "h4", "g4_coprime", "h4_coprime", "f3h3", "f4h4", "num", "denom", "gcd3"]}], "statement": "<p>Evaluate the following sums&nbsp;involving division of fractions. Simplify your answers where possible.&nbsp;</p>", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "fj/gcd1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "fj/gcd1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "gh/gcd1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "gh/gcd1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "f1j1/gcd2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "f1j1/gcd2", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "g1h1/gcd2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "g1h1/gcd2", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle{\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "num/gcd3", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "num/gcd3", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "denom/gcd3", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "denom/gcd3", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} =$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "ad_gcd", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "ad_gcd", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "bcd_gcd", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "bcd_gcd", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}], "tags": ["dividing fractions", "division of fractions", "fractions", "Fractions", "mixed numbers", "taxonomy"], "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n", "js": ""}, "functions": {}, "variables": {"f4h4": {"description": "<p>variable f4 times h4.</p>\n<p>Used in part c)</p>", "group": "part c", "definition": "f4*h4_coprime", "name": "f4h4", "templateType": "anything"}, "g4_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "g4/gcd(g4,h4)", "name": "g4_coprime", "templateType": "anything"}, "h4": {"description": "<p>Random number but not the same number as variable g4.</p>\n<p>Used in part c.</p>", "group": "part c", "definition": "random(5..8 except g4)", "name": "h4", "templateType": "anything"}, "g": {"description": "<p>Random number between 2 and 10 and not the same number as variable f.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(f..12 except f) ", "name": "g", "templateType": "anything"}, "a": {"description": "<p>Random number between 1 and 20</p>\n<p>Used by part d)</p>", "group": "part d", "definition": "random(1..10#1)", "name": "a", "templateType": "randrange"}, "bd_c": {"description": "<p>Unsimplified denominator for part d).</p>", "group": "part d", "definition": "(bd-c)", "name": "bd_c", "templateType": "anything"}, "h3_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "h3/gcd(g3,h3)", "name": "h3_coprime", "templateType": "anything"}, "f_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "f/gcd(f,g)", "name": "f_coprime", "templateType": "anything"}, "g_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "g/gcd(f,g)", "name": "g_coprime", "templateType": "anything"}, "j1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "j1/gcd(h1,j1)", "name": "j1_coprime", "templateType": "anything"}, "gcd2": {"description": "<p>greatest common divisor of variables f1j1 and g1h1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "gcd(f1j1,g1h1)", "name": "gcd2", "templateType": "anything"}, "c": {"description": "<p>Random prime number between -10 and 10.</p>\n<p>Used by part d).</p>", "group": "part d", "definition": "random([-7,-5,-3,-2,-1,1,2,3,5,7] except d)", "name": "c", "templateType": "anything"}, "ad_gcd": {"description": "<p>Correct answer for the numerator in part d)</p>", "group": "part d", "definition": "ad/gcd", "name": "ad_gcd", "templateType": "anything"}, "g1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "g1/gcd(f1,g1)", "name": "g1_coprime", "templateType": "anything"}, "h1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "h1/gcd(h1,j1)", "name": "h1_coprime", "templateType": "anything"}, "gcd3": {"description": "<p>greatest common denominator for part c.&nbsp;</p>", "group": "part c", "definition": "gcd(num,denom)", "name": "gcd3", "templateType": "anything"}, "bd": {"description": "<p>Variable b times variable d.</p>\n<p>Used in part d)</p>", "group": "part d", "definition": "b*d", "name": "bd", "templateType": "anything"}, "j1": {"description": "<p>Random number between 2 and 20 and not the same value as variable h1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(h1..11 except h1)", "name": "j1", "templateType": "anything"}, "g1h1": {"description": "<p>variable g1 times h1.&nbsp;</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "g1_coprime*h1_coprime", "name": "g1h1", "templateType": "anything"}, "f": {"description": "<p>Random number between 2 and 10.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(2..10)", "name": "f", "templateType": "anything"}, "b": {"description": "<p>Random number between 1 and 10.</p>\n<p>Used by part d)</p>", "group": "part d", "definition": "random(1..10#1)", "name": "b", "templateType": "randrange"}, "bcd_gcd": {"description": "<p>Correct answer for the denominator in part d).</p>", "group": "part d", "definition": "{bd_c}/gcd", "name": "bcd_gcd", "templateType": "anything"}, "f4": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3)", "name": "f4", "templateType": "anything"}, "f1": {"description": "<p>Random number between 2 and 20.</p>\n<p>Used in part b)</p>", "group": "part b", "definition": "random(2..10)", "name": "f1", "templateType": "anything"}, "d": {"description": "<p>Random prime number between 10 and 20.</p>\n<p>Used in part d).</p>", "group": "part d", "definition": "random(7,11,13,17)", "name": "d", "templateType": "anything"}, "g3": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3)", "name": "g3", "templateType": "anything"}, "f3h3": {"description": "<p>variable f3 times h3.</p>", "group": "part c", "definition": "f3*h3_coprime", "name": "f3h3", "templateType": "anything"}, "h": {"description": "<p>Random number from 2 to 10.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(2..10)", "name": "h", "templateType": "anything"}, "gh": {"description": "<p>variable g times variable h.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "g_coprime*h_coprime", "name": "gh", "templateType": "anything"}, "j_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "j/gcd(h,j)", "name": "j_coprime", "templateType": "anything"}, "denom": {"description": "<p>Unsimplified denominator of part c.</p>", "group": "part c", "definition": "h3_coprime*(f4h4+g4_coprime)", "name": "denom", "templateType": "anything"}, "j": {"description": "<p>Random number between 2 and 10 and not the same value as h.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(h..12 except h)", "name": "j", "templateType": "anything"}, "f1j1": {"description": "<p>variable f1 times j1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "f1_coprime*j1_coprime", "name": "f1j1", "templateType": "anything"}, "h4_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "h4/gcd(g4,h4)", "name": "h4_coprime", "templateType": "anything"}, "g1": {"description": "<p>Random number between 2 and 30 and not the same value as variable f1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(f1..11 except f1) ", "name": "g1", "templateType": "anything"}, "fj": {"description": "<p>variable f times variable j.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "f_coprime*j_coprime", "name": "fj", "templateType": "anything"}, "gcd": {"description": "<p>Greatest common divisor of ad and bd_c.&nbsp;</p>\n<p>Used in part d).&nbsp;</p>", "group": "part d", "definition": "gcd(ad,bd_c)", "name": "gcd", "templateType": "anything"}, "f3": {"description": "<p>Random number between 2 and 6.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3#1)", "name": "f3", "templateType": "randrange"}, "f1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "f1/gcd(f1,g1)", "name": "f1_coprime", "templateType": "anything"}, "h3": {"description": "<p>Random number and not the same value as variable g3.&nbsp;</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(5..8)", "name": "h3", "templateType": "anything"}, "gcd1": {"description": "<p>greatest common divisor of variable fj and gh.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "gcd(fj,gh)", "name": "gcd1", "templateType": "anything"}, "g3_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "g3/gcd(g3,h3)", "name": "g3_coprime", "templateType": "anything"}, "h_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "h/gcd(h,j)", "name": "h_coprime", "templateType": "anything"}, "g4": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..5)", "name": "g4", "templateType": "anything"}, "h1": {"description": "<p>Random number between 2 and 20.&nbsp;</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(2..10)", "name": "h1", "templateType": "anything"}, "num": {"description": "<p>numerator of the improper fraction in part c. Unsimplified.&nbsp;</p>", "group": "part c", "definition": "h4_coprime*(f3h3+g3_coprime)", "name": "num", "templateType": "anything"}, "ad": {"description": "<p>Variable a times variable d.</p>\n<p>Used in part d).</p>", "group": "part d", "definition": "a*d", "name": "ad", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}, {"name": "Percentages", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Basic percentages Percentages"], "variable_overrides": [[]], "questions": [{"name": "Merryn's copy of Percentages", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["converting", "decimals", "Fractions", "fractions", "percentages"], "metadata": {"description": "<p>&nbsp;Basics, percentage of an amount, converting to fractions and decimals.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..b-1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "mult*random(1..mult2-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random([50,4,25,5,20])", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "mult*mult2", "description": "", "templateType": "anything", "can_override": false}, "nicelist": {"name": "nicelist", "group": "Ungrouped variables", "definition": "shuffle(map(2^a*5^b,[a,b],[[0,1],[0,2],[1,0],[2,0],[3,0],[4,0],[5,0],[6,0],[1,1],[2,1],[3,1],[4,1],[1,2]]))\n", "description": "", "templateType": "anything", "can_override": false}, "factorsofhun": {"name": "factorsofhun", "group": "Ungrouped variables", "definition": "shuffle([[2,50],[4,25],[5,20]])", "description": "", "templateType": "anything", "can_override": false}, "amount2": {"name": "amount2", "group": "Ungrouped variables", "definition": "random(2..12)*factorsofhun[0][1]", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "percent2*amount2/100", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "percent3*amount3/100", "description": "", "templateType": "anything", "can_override": false}, "percent": {"name": "percent", "group": "Ungrouped variables", "definition": "random(1..99 except list(0..100#5))", "description": "", "templateType": "anything", "can_override": false}, "number": {"name": "number", "group": "Ungrouped variables", "definition": "random(12..78)", "description": "", "templateType": "anything", "can_override": false}, "amount3": {"name": "amount3", "group": "Ungrouped variables", "definition": "random(2..12)*factorsofhun[2][1]", "description": "", "templateType": "anything", "can_override": false}, "mult": {"name": "mult", "group": "Ungrouped variables", "definition": "random(3,7,9,11)", "description": "", "templateType": "anything", "can_override": false}, "mult2": {"name": "mult2", "group": "Ungrouped variables", "definition": "random(4,5)", "description": "", "templateType": "anything", "can_override": false}, "percent3": {"name": "percent3", "group": "Ungrouped variables", "definition": "random(2..19)*factorsofhun[2][0]", "description": "", "templateType": "anything", "can_override": false}, "percent2": {"name": "percent2", "group": "Ungrouped variables", "definition": "random(2..19)*factorsofhun[0][0]", "description": "", "templateType": "anything", "can_override": false}, "dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(0.1..0.9#0.01)", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(0.1..0.9#0.001 except list(0.1..0.9#0.01))", "description": "", "templateType": "anything", "can_override": false}, "dec3": {"name": "dec3", "group": "Ungrouped variables", "definition": "random(1..5#0.01)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["percent", "nicelist", "factorsofhun", "amount2", "percent2", "ans2", "amount3", "percent3", "ans3", "b", "a", "mult", "mult2", "c", "d", "number", "dec1", "dec2", "dec3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{percent}% means {percent} out of&nbsp;[[0]].</p>\n<p>This&nbsp;means we can write {percent}% as the fraction [[1]]&nbsp;.</p>\n<p>And so we can write {percent}% as the decimal&nbsp;[[2]].</p>\n<p></p>", "stepsPenalty": "1.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>'Percent' means 'out of 100'. So we can write 34% as $\\frac{34}{100}$. Since this is the same as&nbsp;$34\\div 100$ we can write this as a decimal by moving the decimal point two places (the number of zeroes in 100). So 34 (which has a decimal point after the 4) becomes 0.34. In summary:</p>\n<p>\\[34\\%=\\frac{34}{100}=0.34\\]</p>\n<p>&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "100", "maxValue": "100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{percent}/100", "maxValue": "{percent}/100", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{percent/100}", "maxValue": "{percent/100}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Writing a percentage as a fraction or a decimal can be&nbsp;useful&nbsp;in the following type of questions:</p>\n<p>{percent2}% of {amount2} is&nbsp;[[0]]</p>\n<p>{percent3}% of {amount3} is&nbsp;[[1]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Write the percentage as a fraction or decimal and replace the word 'of' with '$\\times$'. </em></p>\n<p>This question requires you know how to multiply fractions and/or decimals. If using a fraction, it will&nbsp;help to simplify&nbsp;it.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example, 40% of 55 is $40\\%\\times 55=\\frac{40}{100}\\times 55$, which equals $\\frac{2}{5}\\times 55$ by simplifying the fraction, next&nbsp;there is a factor of 5 which can be cancelled to give $2\\times 11=22$. So we have 40% of 55 is 22.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ans2}", "maxValue": "{ans2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ans3}", "maxValue": "{ans3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{a} out of {b} as a fraction is [[0]]. What is this equal to as a percentage?&nbsp;[[1]]% &nbsp;</p>\n<p>{c} out of {d} as a fraction is [[2]]. What is this equal to as a percentage?&nbsp;[[3]]% &nbsp;</p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Recall that $\\frac{2}{3}$ means '2&nbsp;out of 3', and so going the other way, expressions like '4 out of 5' are equivalent to $\\frac{4}{5}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>To convert a fraction into a percentage, make an equivalent fraction with a denominator of 100, then the numerator will be the percentage. For example, given $\\frac{4}{5}$ we can multiply the top and bottom by 20 to get it over 100 (we could have multiplied&nbsp;by 2 and then 10 as well since this is the same thing). Our working could look like this:</p>\n<p>\\[\\frac{4}{5}=\\frac{4\\times 20}{5\\times 20}=\\frac{80}{100}=80\\%\\]&nbsp;</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a}/{b}", "maxValue": "{a}/{b}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a/b*100}", "maxValue": "{a/b*100}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c/d}", "maxValue": "{c/d}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{c/d*100}", "maxValue": "{c/d*100}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>It is often useful to determine 1% of an amount and 10% of an amount so you can combine them to get other percentages. For example</p>\n<p></p>\n<p>10% of {number} is&nbsp;[[0]]</p>\n<p>1% of {number} is&nbsp;[[1]]</p>\n<p>therefore 11% of {number} is &nbsp;[[2]], 9% of {number} is&nbsp;[[3]]&nbsp;and&nbsp;20% of {number} is [[4]].</p>", "stepsPenalty": "2.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Move the decimal point once to the left to get 10%, move it one more time to the left to get 1%. Add and subtract multiples of these as required.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>Notice</p>\n<p>\\[10\\%=\\frac{10}{100}=\\frac{1}{10}=1\\div 10 = 0.1\\]</p>\n<p>and&nbsp;</p>\n<p>\\[1\\%=\\frac{1}{100}=1\\div 100 = 0.01\\]</p>\n<p>This means that 10% of 12 is 1.2 and 1% of 12 is 0.12. From this we can calculate many simple percentages:</p>\n<ul>\n<li>11% by adding&nbsp;10% and 1%, that is $1.2+0.12=1.32$</li>\n<li>9% by doing 10% minus 1%, that is $1.2-0.12=1.08$</li>\n<li>30% by doing 3 times 10%, that is $3\\times 1.2=3.6$</li>\n<li>99% by doing 100% minus&nbsp;1%, that is $12-0.12=11.88$</li>\n</ul>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{number/10}", "maxValue": "{number/10}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{number/100}", "maxValue": "{number/100}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{0.11*number}", "maxValue": "{0.11*number}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{0.09*number}", "maxValue": "{0.09*number}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{0.2*number}", "maxValue": "{0.2*number}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Rewrite the following decimals as percentages:</p>\n<p>{dec1}&nbsp;=&nbsp;[[0]]%</p>\n<p>{dec2} =&nbsp;[[1]]%</p>\n<p>{dec3} =&nbsp;[[2]]%</p>", "stepsPenalty": "1.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>Move the decimal point to the right two places.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>To get from a percentage to a <g class=\"gr_ gr_12 gr-alert gr_gramm Punctuation only-ins replaceWithoutSep\" id=\"12\" data-gr-id=\"12\">decimal</g> we ultimately moved the decimal place twice to the left.&nbsp;Recall&nbsp;</p>\n<p>\\[34\\%=\\frac{34}{100}=34\\div 100=0.34\\]</p>\n<p>That means to go the other way (from a decimal to a percentage) we need to move the decimal place twice to the right, in other words</p>\n<p>\\[0.34=34\\div 100=\\frac{34}{100}=34\\%\\]</p>\n<p>Often it is useful to remember that 1 represents 100%, this can help&nbsp;you to check if your answer makes sense.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dec1*100", "maxValue": "dec1*100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dec2*100", "maxValue": "dec2*100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dec3*100", "maxValue": "dec3*100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Rounding", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Rounding to 0, 1, 2  or 3 decimal places", "Rounding to the nearest 10, 100, 1000"], "variable_overrides": [[], []], "questions": [{"name": "Merryn's copy of Rounding to 0, 1, 2  or 3 decimal places", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Round random numbers to the closest whole number, 1, 2 or 3 decimals places.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"trick": {"name": "trick", "group": "Ungrouped variables", "definition": "random(0.04,0.05, 0.06, 0.07, 0.08)", "description": "", "templateType": "anything", "can_override": false}, "number2": {"name": "number2", "group": "Ungrouped variables", "definition": "random(10.001..300#0.0003)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(number,ndp)", "description": "", "templateType": "anything", "can_override": false}, "digit_to_the_right": {"name": "digit_to_the_right", "group": "Ungrouped variables", "definition": "mod(floor(10^(ndp+1)*number),10)\n", "description": "", "templateType": "anything", "can_override": false}, "number": {"name": "number", "group": "Ungrouped variables", "definition": "random(10.00001..90#0.00002)", "description": "", "templateType": "anything", "can_override": false}, "n4r": {"name": "n4r", "group": "Ungrouped variables", "definition": "random(0.001..0.999#0.001)", "description": "", "templateType": "anything", "can_override": false}, "seed": {"name": "seed", "group": "Ungrouped variables", "definition": "random(1,0.1,0.01,0.001)", "description": "", "templateType": "anything", "can_override": false}, "ndp": {"name": "ndp", "group": "Ungrouped variables", "definition": "countdp(string(seed))", "description": "", "templateType": "anything", "can_override": false}, "direction": {"name": "direction", "group": "Ungrouped variables", "definition": "if(digit_to_the_right<5,'down', 'up')", "description": "", "templateType": "anything", "can_override": false}, "less_greater": {"name": "less_greater", "group": "Ungrouped variables", "definition": "if(digit_to_the_right<5,'less than', 'greater than or equal to')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["seed", "ndp", "number", "digit_to_the_right", "ans", "direction", "less_greater", "number2", "trick", "n4r"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{number}$ rounded to <span data-jme-visible=\"ndp=0\">the nearest whole number</span> <span data-jme-visible=\"ndp=1\">{ndp} decimal place</span> <span data-jme-visible=\"ndp=2 or ndp=3\">{ndp} decimal places</span>&nbsp;is [[0]].</p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>When we are rounding we look at the first digit that we might discard. If it is $5$ or greater we round up. If it is less than $5$ we round down.</p>\n<p>We want to round to <span data-jme-visible=\"ndp=0\">the nearest whole number</span> <span data-jme-visible=\"ndp=1\">{ndp} decimal place</span> <span data-jme-visible=\"ndp=2 or ndp=3\">{ndp} decimal places</span>. The digit to the right is $\\var{digit_to_the_right}$, which is {less_greater} $5$ so we round {direction} to $\\var{ans}$&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Rounding to the nearest 10, 100, 1000", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>A single question which asks you to round to the nearest ten, hundred or thousand.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"number": {"name": "number", "group": "Ungrouped variables", "definition": "random(10*seed..50*seed except 10*seed..50*seed#seed)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "tonearest(number, seed)\n", "description": "", "templateType": "anything", "can_override": false}, "digit_to_the_right": {"name": "digit_to_the_right", "group": "Ungrouped variables", "definition": "mod(floor(10*number/seed),10)\n//(mod(number,seed)-mod(number, seed/10))/(seed/10)\n//floor(lastdigits/(seed/10))", "description": "", "templateType": "anything", "can_override": false}, "seed": {"name": "seed", "group": "Ungrouped variables", "definition": "random(10,100,1000)", "description": "", "templateType": "anything", "can_override": false}, "direction": {"name": "direction", "group": "Ungrouped variables", "definition": "if(digit_to_the_right<5,'down', 'up')", "description": "", "templateType": "anything", "can_override": false}, "seedtext": {"name": "seedtext", "group": "Ungrouped variables", "definition": "switch(seed=10, 'ten', seed=100, 'hundred', seed=1000, 'thousand', 'error')", "description": "", "templateType": "anything", "can_override": false}, "less_greater": {"name": "less_greater", "group": "Ungrouped variables", "definition": "if(digit_to_the_right<5,'less than', 'greater than or equal to')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["seed", "seedtext", "number", "digit_to_the_right", "direction", "less_greater", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{number}$ rounded to the nearest {seedtext} is [[0]].</p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>When we are rounding we look at the first digit that we might discard. If it is $5$ or greater we round up. If it is less than $5$ we round down.</p>\n<p></p>\n<p>To round $\\var{number}$ to the nearest {seedtext}, we look to the right of the {seedtext}s column and see the digit $\\var{digit_to_the_right}$.&nbsp;</p>\n<p>Since $\\var{digit_to_the_right}$ is {less_greater} $5$ we round {direction} to $\\var{ans}$.&nbsp;</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Scientific notation", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Scientific notation: large numbers to scientific notation", "Scientific notation: scientific notation to large numbers", "Scientific notation: scientific notation to small numbers", "Scientific notation: small numbers to scientific notation"], "variable_overrides": [[], [], [], []], "questions": [{"name": "Merryn's copy of Scientific notation: large numbers to scientific notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["converting", "scientific notation", "standard form"], "metadata": {"description": "<p>convert large numbers to scientific notation</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>\n<p></p>\n<p>Write the following numbers in scientific notation.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pow2": {"name": "pow2", "group": "Ungrouped variables", "definition": "random(4..8)", "description": "", "templateType": "anything", "can_override": false}, "pow1": {"name": "pow1", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "precround(dec2*10^pow2,0)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "precround(dec1*10^pow1,0)", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.001)", "description": "", "templateType": "anything", "can_override": false}, "dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.01)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec1", "pow1", "q1", "dec2", "pow2", "q2"], "variable_groups": [], "functions": {"spacenumber": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "definition": "var parts=n.toString().split(\".\");\n        if(parts[1] && parts[1].length<2) {\n      parts[1]+='0';\n        }\n        return parts[0].replace(/\\B(?=(\\d{3})+(?!\\d))/g, \" \") + (parts[1] ? \", \" + parts[1] : \"\");"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{q1} =$&nbsp;[[0]]$\\times 10$&nbsp;<sup>[[1]]</sup></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]&nbsp;</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Suppose we have the number $\\var{q1}$. In scientific notation, this number would start with $\\var{dec1}$ since we only want one digit in front of the decimal point. The decimal point&nbsp;is&nbsp;currently to the right of the last digit&nbsp;in $\\var{q1}$ and needs to move to between the first and second digits, that is $\\var{dec1}$. Count the places that the decimal point must jump and you get $\\var{pow1}$ places. That is,</p>\n<p></p>\n<p>\\[\\var{q1}=\\var{dec1}\\times 10^{\\var{pow1}}\\]</p>\n<p></p>\n<p>We have a <em>positive&nbsp;$\\var{pow1}$</em>&nbsp;as the power because we need to make the number $\\var{dec1}$ <em>bigger</em> to get to $\\var{q1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{dec1}", "maxValue": "{dec1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{pow1}", "maxValue": "{pow1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{q2} =$&nbsp;[[0]]$\\times 10$&nbsp;<sup>[[1]]&nbsp;</sup></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]&nbsp;</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Suppose we have the number $\\var{q2}$. In scientific notation, this number would start with $\\var{dec2}$ since we only want one digit in front of the decimal point.&nbsp;The decimal point&nbsp;is&nbsp;currently to the right of the last digit&nbsp;in&nbsp;$\\var{q2}$ and needs to move to between the first and second digits, that is $\\var{dec2}$. Count the places that the decimal point must jump and you get $\\var{pow2}$ places. That is,</p>\n<p></p>\n<p>\\[\\var{q2}=\\var{dec2}\\times 10^{\\var{pow2}}\\]</p>\n<p></p>\n<p>We have a&nbsp;<em>positive&nbsp;$\\var{pow2}$</em>&nbsp;as the power because we need to make the number $\\var{dec2}$&nbsp;<em>bigger</em>&nbsp;to get to $\\var{q2}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{dec2}", "maxValue": "{dec2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{pow2}", "maxValue": "{pow2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Scientific notation: scientific notation to large numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["converting", "scientific notation", "standard form"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>\n<p></p>\n<p>Write the following numbers in decimal&nbsp;notation.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.01)", "description": "", "templateType": "anything", "can_override": false}, "pow2": {"name": "pow2", "group": "Ungrouped variables", "definition": "random(4..8 except pow1)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "precround(dec1*10^pow1,0)", "description": "", "templateType": "anything", "can_override": false}, "pow1": {"name": "pow1", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.001)", "description": "", "templateType": "anything", "can_override": false}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "precround(dec2*10^pow2,0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec1", "pow1", "q1", "dec2", "pow2", "q2"], "variable_groups": [], "functions": {"spacenumber": {"parameters": [["n", "number"]], "type": "string", "language": "javascript", "definition": "var parts=n.toString().split(\".\");\n        if(parts[1] && parts[1].length<2) {\n      parts[1]+='0';\n        }\n        return parts[0].replace(/\\B(?=(\\d{3})+(?!\\d))/g, \" \") + (parts[1] ? \", \" + parts[1] : \"\");"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec1}\\times 10^\\var{pow1}$&nbsp;=&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Recall that multiplying by powers of 10 moves the decimal point, for example multiplying a number by $10^\\var{pow1}$ (which is the same as multiplying by $\\var{10^pow1}$) moves the decimal point&nbsp;$\\var{pow1}$ places to make the number bigger (the decimal point moves to the right). In particular:</p>\n<p>\\[\\var{dec1}\\times 10^\\var{pow1}= \\var{q1}\\]</p>\n<p><em>Note there is a decimal point after the last zero that we do not write simply because there is no reason to.</em></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q1}", "maxValue": "{q1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec2}\\times 10^\\var{pow2}$&nbsp;=&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Recall that multiplying by powers of 10 moves the decimal point, for example multiplying a number by $10^\\var{pow2}$ (which is the same as multiplying by $\\var{10^pow2}$) moves the decimal point&nbsp;$\\var{pow2}$ places to make the number bigger (the decimal point moves to the right). In particular:</p>\n<p>\\[\\var{dec2}\\times 10^\\var{pow2}= \\var{q2}\\]</p>\n<p><em>Note there is a decimal point after the last zero that we do not write simply because there is no reason to.</em></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q2}", "maxValue": "{q2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Scientific notation: scientific notation to small numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["converting", "scientific notation", "standard form"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>\n<p></p>\n<p>Write the following numbers in decimal&nbsp;notation.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q2": {"name": "q2", "group": "Ungrouped variables", "definition": "precround(dec2*10^pow2,14)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "precround(dec1*10^pow1,14)", "description": "", "templateType": "anything", "can_override": false}, "pow2": {"name": "pow2", "group": "Ungrouped variables", "definition": "random(list(-6..-1) except pow1)", "description": "", "templateType": "anything", "can_override": false}, "dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.01)", "description": "", "templateType": "anything", "can_override": false}, "pow1": {"name": "pow1", "group": "Ungrouped variables", "definition": "random(-1..-6)", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.001)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec1", "pow1", "q1", "dec2", "pow2", "q2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec1}\\times 10^\\var{pow1}$ =&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Multiplying a number by $10^\\var{pow1}$ will move the decimal point&nbsp;$\\var{-pow1}$ place<span data-jme-visible=\"pow1&lt;&gt;-1\">s</span> to make the number smaller (the decimal point moves to the left). This is because</p>\n<p>\\[10^\\var{pow1}=\\frac{1}{10^\\var{-pow1}}=\\frac{1}{\\var{10^-pow1}}=\\var{10^pow1}\\]</p>\n<p>and so multiplying by it must make the original number smaller. In particular,</p>\n<p>\\[\\var{dec1}\\times 10^\\var{pow1}=\\var{q1}\\]</p>\n<p><em>Note you should always put a zero in front of your decimal point.</em></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q1}", "maxValue": "{q1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{dec2}\\times 10^\\var{pow2}$ =&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]</p>\n<p>are both in scientific notation.</p>\n<p></p>\n<hr/>\n<p>Multiplying a number by $10^\\var{pow2}$ will move the decimal point&nbsp;$\\var{-pow2}$ place<span data-jme-visible=\"pow2&lt;&gt;-1\">s</span> to make the number smaller (the decimal point moves to the left). This is because</p>\n<p>\\[10^\\var{pow2}=\\frac{1}{10^\\var{-pow2}}=\\frac{1}{\\var{10^-pow2}}=\\var{10^pow2}\\]</p>\n<p>and so multiplying by it must make the original number smaller. In particular,</p>\n<p>\\[\\var{dec2}\\times 10^\\var{pow2}=\\var{q2}\\]</p>\n<p><em>Note you should always put a zero in front of your decimal point.</em></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q2}", "maxValue": "{q2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Scientific notation: small numbers to scientific notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["converting", "scientific notation", "standard form"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>\n<p></p>\n<p>Write the following numbers in scientific notation.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pow2": {"name": "pow2", "group": "Ungrouped variables", "definition": "random(list(-6..-1) except pow1)", "description": "", "templateType": "anything", "can_override": false}, "dec2": {"name": "dec2", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.001)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "precround(dec1*10^pow1,14)", "description": "", "templateType": "anything", "can_override": false}, "dec1": {"name": "dec1", "group": "Ungrouped variables", "definition": "random(1.1..9.9#0.01)", "description": "", "templateType": "anything", "can_override": false}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "precround(dec2*10^pow2,14)", "description": "", "templateType": "anything", "can_override": false}, "pow1": {"name": "pow1", "group": "Ungrouped variables", "definition": "random(-1..-6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec1", "pow1", "q1", "dec2", "pow2", "q2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{q1}$ =&nbsp;[[0]]$\\times 10$&nbsp;<sup>[[1]]</sup></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]&nbsp;</p>\n<p>are both in scientific notation.</p>\n<hr/>\n<p>Suppose we have the number $\\var{q1}$. In scientific notation, this number would start with $\\var{dec1}$ since we only want one digit in front of the decimal point. The decimal point&nbsp;is&nbsp;currently here $\\var{q1}$ and needs to move to between the&nbsp;first and second digits, that is $\\var{dec1}$. Count the places that the decimal point must jump and you get $\\var{-pow1}$ places. That is,&nbsp;&nbsp;</p>\n<p></p>\n<p>\\[\\var{q1}=\\var{dec1}\\times 10^{\\var{pow1}}\\]</p>\n<p></p>\n<p>We have a <em>negative $\\var{-pow1}$</em>&nbsp;as the power because we need to make the number $\\var{dec1}$ <em>smaller</em> to get to $\\var{q1}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{dec1}", "maxValue": "{dec1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{pow1}", "maxValue": "{pow1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{q2}$ =&nbsp;[[0]]$\\times 10$&nbsp;<sup>[[1]]</sup></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>A number is in scientific notation if it is written as a decimal multiplied by some power of 10, where the decimal has exactly one digit in front of the decimal place. For example:</p>\n<p>\\[1.234\\times 10^6, \\quad \\text{and} \\quad 3.01\\times 10^{-3}\\]&nbsp;</p>\n<p>are both in scientific notation.</p>\n<hr/>\n<p>Suppose we have the number $\\var{q2}$. In scientific notation, this number would start with $\\var{dec2}$ since we only want one digit in front of the decimal point.&nbsp;The decimal point&nbsp;is&nbsp;currently here $\\var{q2}$ and needs to move to between the&nbsp;first and second digits, that is $\\var{dec2}$. Count the places that the decimal point must jump and you get $\\var{-pow2}$ places. That is,&nbsp;&nbsp;</p>\n<p></p>\n<p>\\[\\var{q2}=\\var{dec2}\\times 10^{\\var{pow2}}\\]</p>\n<p></p>\n<p>We have a&nbsp;<em>negative&nbsp;$\\var{-pow2}$</em>&nbsp;as the power because we need to make the number $\\var{dec2}$&nbsp;<em>smaller</em>&nbsp;to get to $\\var{q2}$.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{dec2}", "maxValue": "{dec2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{pow2}", "maxValue": "{pow2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Unit conversions", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [" Units: common units", " Units: prefixes", "Units: converting between different prefixes"], "variable_overrides": [[], [], []], "questions": [{"name": "Merryn's copy of Units: common units", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["capacity", "distance", "Distance", "length", "Time", "time", "units", "volume", "Volume", "weight"], "metadata": {"description": "<p>Recall of common units, along with understanding multiplication.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>In the second sentence&nbsp;for each part&nbsp;you can enter multiplication using&nbsp;an asterisk, for example $60\\times 12$ can be written as 60*12.</p>\n<p>Avoid using commas or spaces between zeroes, for example&nbsp;ten thousand would be entered 10000.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"distance": {"name": "distance", "group": "Ungrouped variables", "definition": "random([\"millimetres\",\"kilometre\", 1000000], [\"centimetres\", \"kilometre\",100000],[\"millimetres\",\"metre\",1000])", "description": "", "templateType": "anything", "can_override": false}, "capacity": {"name": "capacity", "group": "Ungrouped variables", "definition": "random([\"millilitres\",\"kilolitre\",1000000],[\"millilitres\",\"megalitre\",1000000000])", "description": "", "templateType": "anything", "can_override": false}, "mass": {"name": "mass", "group": "Ungrouped variables", "definition": "random([\"grams\",\"tonne\",1000000],[\"milligrams\",\"tonne\",1000000000],[\"milligrams\",\"kilogram\",1000000])", "description": "", "templateType": "anything", "can_override": false}, "time": {"name": "time", "group": "Ungrouped variables", "definition": "random([\"milliseconds\",\"hour\",3600000],[\"minutes\",\"day\",1440],[\"seconds\",\"day\",86400],[\"milliseconds\",\"day\",24*3600000])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["distance", "time", "mass", "capacity"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are&nbsp;[[0]]&nbsp;millimetres in a centimetre, [[1]]&nbsp;centimetres in a metre and&nbsp;[[2]] metres in a kilometre.</p>\n<p>From this we can calculate that there are [[3]]&nbsp;{distance[0]}&nbsp;in a {distance[1]}.</p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Suppose you knew there were 100 centimetres in a metre and 1000 metres in a kilometre.</p>\n<p>To determine how many centimetres are in a kilometre think \"there are 1000 metres in a kilometre, and each metre has 100 centimetres in it, therefore there are $1000\\times 100$ centimetres in a kilometre\".&nbsp;</p>\n<p>You would normally write this as&nbsp;100 000 but here you need to remove the space or write it as a multiplication, for example, 1000*100.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "10", "maxValue": "10", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "100", "maxValue": "100", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{distance[2]}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are&nbsp;[[0]]&nbsp;milliseconds in a second,&nbsp;[[1]] seconds in a minute,&nbsp;[[2]] minutes in an hour, and&nbsp;[[3]] hours in a day.</p>\n<p>From this we can calculate that&nbsp;there are [[4]]&nbsp;{time[0]} in a {time[1]}.</p>", "stepsPenalty": "2.5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Suppose you knew there were 1000&nbsp;milliseconds&nbsp;in a second&nbsp;and 60 seconds in a minute.</p>\n<p>To determine how many milliseconds are in a minute think \"there are 60 seconds in a minute, and each second&nbsp;has 1000 milliseconds in it, therefore there are $60\\times 1000$ milliseconds in a minute\".&nbsp;</p>\n<p>You would normally write this as 60 000 but here you need to remove the space or write it as a multiplication, for example, 60*1000.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "60", "maxValue": "60", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "60", "maxValue": "60", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "24", "maxValue": "24", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{time[2]}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are&nbsp;[[0]] milligrams in a gram, [[1]] grams in a kilogram,&nbsp;[[2]] kilograms in a tonne.&nbsp;</p>\n<p>From this we can calculate that&nbsp;there are [[3]]&nbsp;{mass[0]} in a {mass[1]}.</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Suppose you knew there were 1000&nbsp;milligrams in a gram&nbsp;and 1000&nbsp;grams in a kilogram.</p>\n<p>To determine how many milligrams are in a kilogram&nbsp;think \"there are 1000&nbsp;grams in a kilogram, and each gram&nbsp;has 1000 milligrams in it, therefore there are $1000\\times 1000$ milligrams in a kilogram\".&nbsp;</p>\n<p>You would normally write this as 1 000 000 but here you need to remove the space or write it as a multiplication, for example, 1000*1000.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{mass[2]}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There are [[0]]&nbsp;millilitres in a litre, [[1]]&nbsp;litres in a kilolitre and [[2]]&nbsp;litres in a megalitre.</p>\n<p>From this we can calculate that there are [[3]]&nbsp;{capacity[0]}&nbsp;in a {capacity[1]}.</p>", "stepsPenalty": "2", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Suppose you knew there were 1000&nbsp;millilitres in a litre&nbsp;and 1 000 000 litres&nbsp;in a megalitre.</p>\n<p>To determine how many millilitres are in a megalitre&nbsp;think \"there are 1 000 000 litres&nbsp;in a megalitre, and each litre&nbsp;has 1000 millilitres in it, therefore there are $1\\,000\\,000\\times&nbsp;1000 $ millilitres&nbsp;in a megalitre\".&nbsp;</p>\n<p>You would normally write this as 1 000 000 000 but here you need to remove the space or write it as a multiplication, for example, 1000000*1000.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000", "maxValue": "1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1000000", "maxValue": "1000000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{capacity[2]}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Units: prefixes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": ["prefix", "prefixes", "SI", "units"], "metadata": {"description": "<p>Match prefixes with their abbreviations, from nano to giga. Match prefixes with equivalent scientific notation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<table class=\"infobox\" cellpadding=\"1\">\n<tbody>\n<tr>\n<th colspan=\"4\">Metric prefixes in everyday use</th>\n</tr>\n<tr>\n<th>Text</th>\n<th>Symbol</th>\n<th>Multiple of the base unit</th>\n<th>Multiple of the base unit (as a power)</th>\n</tr>\n<tr>\n<td><g class=\"gr_ gr_163 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"163\" data-gr-id=\"163\">tera</g></td>\n<td>T</td>\n<td><span class=\"nowrap\">1000000000000</span></td>\n<td>10<sup>12</sup></td>\n</tr>\n<tr>\n<td>giga</td>\n<td>G</td>\n<td><span class=\"nowrap\">1000000000</span></td>\n<td>10<sup>9</sup></td>\n</tr>\n<tr>\n<td>mega</td>\n<td>M</td>\n<td><span class=\"nowrap\">1000000</span></td>\n<td>10<sup>6</sup></td>\n</tr>\n<tr>\n<td>kilo</td>\n<td>k</td>\n<td><span class=\"nowrap\">1000</span></td>\n<td>10<sup>3</sup></td>\n</tr>\n<tr>\n<td><g class=\"gr_ gr_164 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"164\" data-gr-id=\"164\">hecto</g></td>\n<td>h</td>\n<td>100</td>\n<td>10<sup>2</sup></td>\n</tr>\n<tr>\n<td>deca</td>\n<td>da</td>\n<td>10</td>\n<td>10<sup>1</sup></td>\n</tr>\n<tr>\n<td><g class=\"gr_ gr_165 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"165\" data-gr-id=\"165\">deci</g></td>\n<td>d</td>\n<td>0.1</td>\n<td>10<sup>&minus;1</sup></td>\n</tr>\n<tr>\n<td><g class=\"gr_ gr_166 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"166\" data-gr-id=\"166\">centi</g></td>\n<td>c</td>\n<td>0.01</td>\n<td>10<sup>&minus;2</sup></td>\n</tr>\n<tr>\n<td>milli</td>\n<td>m</td>\n<td>0.001</td>\n<td>10<sup>&minus;3</sup></td>\n</tr>\n<tr>\n<td>micro</td>\n<td>mc or $\\mu$</td>\n<td><span class=\"nowrap\">0.000001</span></td>\n<td>10<sup>&minus;6</sup></td>\n</tr>\n<tr>\n<td>nano</td>\n<td>n</td>\n<td><span class=\"nowrap\">0.000000001</span></td>\n<td>10<sup>&minus;9</sup></td>\n</tr>\n<tr>\n<td>pico</td>\n<td>p</td>\n<td><span class=\"nowrap\">0.000000000001</span></td>\n<td>10<sup>&minus;12</sup></td>\n</tr>\n</tbody>\n</table>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>For each prefix listed on the left click the button that corresponds to the correct abbreviation listed above.</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["nano", "micro", "milli", "centi", "deci", "kilo", "mega", "giga"], "matrix": [["1", 0, 0, 0, 0, 0, 0, 0], ["0", "1", 0, 0, 0, 0, 0, 0], [0, 0, "1", 0, 0, 0, 0, 0], [0, 0, 0, "1", 0, 0, 0, 0], [0, 0, 0, 0, "1", 0, 0, 0], [0, 0, 0, 0, 0, "1", 0, 0], [0, 0, 0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["n", "mc or $\\mu$", "m", "c", "d", "k", "M", "G"]}, {"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>For each prefix listed on the left click the button that corresponds to&nbsp;how many of the base unit it represents.</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["nano", "micro", "milli", "centi", "centi", "deci", "kilo", "mega", "giga"], "matrix": [["1", "0", 0, 0, 0, 0, 0, 0, 0], ["0", "1", "0", 0, 0, 0, 0, 0, 0], [0, 0, "1", "0", 0, 0, 0, 0, 0], [0, 0, 0, "1", 0, 0, 0, 0, 0], [0, 0, 0, 0, "1", 0, 0, 0, 0], [0, 0, 0, 0, 0, "1", 0, 0, 0], [0, 0, 0, 0, 0, 0, "1", 0, 0], [0, 0, 0, 0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, 0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["$10^{-9}$", "$10^{-6}$", "$10^{-3}$", "$10^{-2}$", "$10^{-1}$", "$10^{1}$", "$10^{3}$", "$10^{6}$", "$10^{9}$"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Merryn's copy of Units: converting between different prefixes 1D (nano to giga)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Converting between giga, mega, kilo, base, milli and micro, nano. Metres, grams and litres.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following questions down on paper and evaluate them without using a calculator using any method you choose.</p>\n<p>If you click on <em>Show steps you will see</em> full working using the standard method. Click on <em>Try another question like this one</em> to get a new pair of numbers.</p>", "advice": "<p>Write the following questions down on paper and evaluate&nbsp;them without using a calculator.</p>\n<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Show steps</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"random_order": {"name": "random_order", "group": "Ungrouped variables", "definition": "shuffle(0..len(prefix)-1)", "description": "", "templateType": "anything", "can_override": false}, "apowerdiff": {"name": "apowerdiff", "group": "Ungrouped variables", "definition": "P1[3]-P0[3]", "description": "", "templateType": "anything", "can_override": false}, "ansa": {"name": "ansa", "group": "Ungrouped variables", "definition": "precround(mult[0]*10^(P0[3]-P1[3]),P1[3]-P0[3]+2)", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "shuffle([[\"L\",\"litres\"],[\"m\",\"metres\"],[\"g\",\"grams\"]])[0..2]", "description": "", "templateType": "anything", "can_override": false}, "P2": {"name": "P2", "group": "Ungrouped variables", "definition": "prefix[RP2[0]]", "description": "", "templateType": "anything", "can_override": false}, "P3": {"name": "P3", "group": "Ungrouped variables", "definition": "prefix[RP2[1]]", "description": "", "templateType": "anything", "can_override": false}, "RP1": {"name": "RP1", "group": "Ungrouped variables", "definition": "sort(random_order[0..2])", "description": "<p>random pair 1 - designed to be increasing</p>", "templateType": "anything", "can_override": false}, "ansb": {"name": "ansb", "group": "Ungrouped variables", "definition": "precround(mult[1]*10^(P2[3]-P3[3]),0)", "description": "", "templateType": "anything", "can_override": false}, "P1": {"name": "P1", "group": "Ungrouped variables", "definition": "prefix[RP1[1]]", "description": "", "templateType": "anything", "can_override": false}, "prefix": {"name": "prefix", "group": "Ungrouped variables", "definition": "[[\"n\",10^(-9),\"nano\",-9],[\"mc\",10^(-6),\"micro\",-6],[\"m\",10^(-3),\"milli\",-3],[\"\",1,\"the base unit, \",0],[\"k\",10^3,\"kilo\",3],[\"M\",10^6,\"mega\",6],[\"G\",10^9,\"giga\",9]]", "description": "", "templateType": "anything", "can_override": false}, "mult": {"name": "mult", "group": "Ungrouped variables", "definition": "shuffle([random(3..191),random(0.1..2.9#0.01 except [1,2])])", "description": "", "templateType": "anything", "can_override": false}, "thousandsa": {"name": "thousandsa", "group": "Ungrouped variables", "definition": "abs(P0[3]-P1[3])/3", "description": "", "templateType": "anything", "can_override": false}, "RP2": {"name": "RP2", "group": "Ungrouped variables", "definition": "if(random_order[2]>random_order[3],random_order[2..4],[random_order[3],random_order[2]])", "description": "<p>random pair 2 - designed to be decreasing&nbsp;</p>", "templateType": "anything", "can_override": false}, "bpowerdiff": {"name": "bpowerdiff", "group": "Ungrouped variables", "definition": "P2[3]-P3[3]", "description": "", "templateType": "anything", "can_override": false}, "P0": {"name": "P0", "group": "Ungrouped variables", "definition": "prefix[RP1[0]]", "description": "", "templateType": "anything", "can_override": false}, "thousandsb": {"name": "thousandsb", "group": "Ungrouped variables", "definition": "(P2[3]-P3[3])/3", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["prefix", "units", "mult", "random_order", "RP1", "RP2", "P0", "P1", "P2", "P3", "ansa", "apowerdiff", "ansb", "bpowerdiff", "thousandsa", "thousandsb"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{mult[0]}\\, \\var{P0[0]}\\var{units[0][0]} =$&nbsp;[[0]]&nbsp;$\\var{P1[0]}\\var{units[0][0]}$&nbsp;</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The following are some of the common SI prefixes used listed in descending order. The scaling factor between each adjacent prefix is $10^3=1000$.&nbsp;</p>\n<p>giga<br/>mega<br/>kilo<br/>base unit<br/>milli<br/>micro<br/>nano</p>\n<p></p>\n<p>The scaling factor between {P0[2]}&nbsp;and {P1[2]}&nbsp;is therefore</p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"thousandsa&lt;&gt;1\">$(1000)^\\var{thousandsa}=$</span>$10^\\var{apowerdiff}=\\simplify{10^{apowerdiff}}$.</p>\n<p style=\"text-align: center;\"></p>\n<p>When converting from&nbsp;{P0[2]+units[0][1]} to {P1[2]+units[0][1]} the <em>units</em>&nbsp;are getting <em>larger</em>&nbsp;and so the the <em>number</em> will have to get <em>smaller</em>. That is, we will <em>divide</em> by $\\simplify{10^{apowerdiff}}$ to convert from&nbsp;{P0[2]+units[0][1]} to {P1[2]+units[0][1]}. Recall <em>dividing</em> by&nbsp;$\\simplify{10^{apowerdiff}}$ is simply moving the decimal place to the <em>left</em> $\\var{apowerdiff}$ places in order to make the number <em>smaller</em>.</p>\n<p></p>\n<p>Therefore, \\begin{align}\\var{mult[0]} \\,\\var{P0[0]}\\var{units[0][0]}&amp;=\\left(\\var{mult[0]} \\div\\simplify{10^{P1[3]-P0[3]}} \\right)\\,\\var{P1[0]}\\var{units[0][0]}\\\\&amp;=\\var{ansa}\\,\\var{P1[0]}\\var{units[0][0]}.\\end{align}</p>\n<p></p>\n<p>If this doesn't make sense to you consider a simpler example. $5$ one-dollar coins is the same amount of money as $1$ five-dollar note. As the units get bigger you need less of them to have the same amount of money. In this example, the scaling factor would be $5$ since the size of the units increases by a factor of $5$ and the number of units decreases by a factor of $5$. Now, say we had $20$ one-dollar coins and we wanted to convert this to five-dollar notes, we would calculate $20\\div 5$ which equals $4$ and therefore we would get $4$ five-dollar notes.&nbsp;</p>\n<p></p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ansa}", "maxValue": "{ansa}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{mult[1]}\\, \\var{P2[0]}\\var{units[1][0]} =$&nbsp;[[0]]&nbsp;$\\var{P3[0]}\\var{units[1][0]}$</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The following are some of the common SI prefixes used list in decending order. The scaling factor between each adjacent prefix is $10^3=1000$.&nbsp;</p>\n<p>giga<br/>mega<br/>kilo<br/>base unit<br/>milli<br/>micro<br/>nano</p>\n<p></p>\n<p>The scaling factor between {P2[2]}&nbsp;and {P3[2]}&nbsp;is&nbsp; therefore</p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"thousandsb&lt;&gt;1\">$(10^3)^\\var{thousandsb}=$</span>$10^\\var{bpowerdiff}=\\simplify{10^{bpowerdiff}}$.</p>\n<p>When converting from&nbsp;{P2[2]+units[1][1]} to {P3[2]+units[1][1]} the&nbsp;<em>units</em>&nbsp;are getting&nbsp;<em>smaller</em>&nbsp;and so the the&nbsp;<em>number</em>&nbsp;will have to get&nbsp;<em>bigger</em>. That is, we will&nbsp;<em>multiply</em> by $\\simplify{10^{bpowerdiff}}$ to convert from&nbsp;{P2[2]+units[1][1]} to {P3[2]+units[1][1]}. Recall&nbsp;<em>multiplying</em>&nbsp;by&nbsp;$\\simplify{10^{bpowerdiff}}$ is simply moving the decimal place to the&nbsp;<em>right</em>&nbsp;$\\var{bpowerdiff}$ places in order to make the number <em>larger</em>.</p>\n<p></p>\n<p>Therefore, \\begin{align}\\var{mult[1]} \\,\\var{P2[0]}\\var{units[1][0]}&amp;=\\left(\\var{mult[1]} \\times\\simplify{10^{bpowerdiff}} \\right)\\,\\var{P3[0]}\\var{units[1][0]}\\\\&amp;=\\var{ansb}\\,\\var{P3[0]}\\var{units[1][0]}.\\end{align}</p>\n<p></p>\n<p>If this doesn't make sense to you consider a simpler example.&nbsp;$1$ five-dollar note is the same amount of money as $5$ one-dollar coins. As the units get&nbsp;smaller you need&nbsp;more of them to have the same amount of money. In this example the scaling factor would&nbsp;be $5$ since the size of the units&nbsp;decreases by a factor of $5$ and the number of units&nbsp;increases by a factor of $5$. Now, say we had $4$ five-dollar notes and we wanted to convert this to one-dollar coins, we would&nbsp;calculate&nbsp;$4\\times&nbsp;5$ which equals $20$&nbsp;and therefore we would get $20$ one-dollar coins.&nbsp;</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ansb}", "maxValue": "{ansb}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Perimeter and area", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["Rectangle area and perimeter", "Finding area or perimeter of a rectangle involving unit conversion", "Area and perimeter of triangles", "Circle area and circumference", "Rectangle Area and Perimeter Challenge problem"], "variable_overrides": [[], [], [], [], []], "questions": [{"name": "Merryn's copy of Rectangle area and perimeter", "extensions": [], "custom_part_types": [], "resources": [["question-resources/rectangle_fFSg27v.png", "/srv/numbas/media/question-resources/rectangle_fFSg27v.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Rachel Staddon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/901/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "functions": {}, "ungrouped_variables": ["rect", "a", "b", "perimeter", "area"], "tags": [], "advice": "<p>The perimeter is the lengths of all the sides added together.</p>\n<p>The area of a rectangle is height x base.</p>", "rulesets": {}, "parts": [{"integerPartialCredit": 0, "prompt": "<p>What is the perimeter of the rectangle in cm?</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{perimeter}", "minValue": "{perimeter}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "prompt": "<p>What is the area of the rectangle in cm$^2$?</p>", "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{area}", "minValue": "{area}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>The diagram below shows a rectangle.</p>\n<p>{image('resources/question-resources/'+rect)}</p>\n<p>a = {a} cm</p>\n<p>b = {b} cm</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "perimeter": {"definition": "a+a+b+b", "templateType": "anything", "group": "Ungrouped variables", "name": "perimeter", "description": ""}, "b": {"definition": "random(10..30)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "rect": {"definition": "'rectangle_fFSg27v.png'", "templateType": "anything", "group": "Ungrouped variables", "name": "rect", "description": ""}, "area": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "area", "description": ""}}, "metadata": {"description": "<p>Area and perimeter of rectangles</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Merryn's copy of Formula for area or perimeter", "extensions": ["random_person"], "custom_part_types": [], "resources": [["question-resources/handball_court.png", "/srv/numbas/media/question-resources/handball_court.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Adelle Colbourn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2083/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>The dimensions of a rectangle are given in m and cm. Compute the area or perimeter.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>{name1} and {name2} were asked by their teacher to measure the school's rectangular handball court and work out the {quantity}.</p>\n<p>{name1} measured the length in metres, but {name2} measured the width in centimetres.</p>\n<p><img src=\"resources/question-resources/handball_court.png\" width=\"659\" height=\"340\"/></p>\n<p></p>", "advice": "<p>{advice}</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"scenarios": {"name": "scenarios", "group": "Ungrouped variables", "definition": "json_decode(safe(\"[\\n{\\n\\\"qn\\\":\\\"Which formula can be applied to their measurements to correctly give the area of the handball court in square metres?\\\",\\n\\\"quant\\\":\\\"area\\\",\\n\\\"ans\\\":\\\"Area = length x (width \u00f7 100)\\\",\\n\\\"dis1\\\":\\\"Area = length x width\\\",\\n\\\"dis2\\\":\\\"Area = length x (width x 100)\\\",\\n\\\"dis3\\\":\\\"Area = 2 x length + 2 x width \u00f7 100\\\",\\n\\\"adv\\\":\\\"The answer is to be given in square metres, so the width needs to be converted from centimetres to metres by dividing by 100.<br>The area of any rectangle is found by multiplying the length by the width.<br>So Area = length x (width \u00f7 100) is the correct formula.<br>Note that multiplication and division have equal priority in the order of operations, so the position of the brackets does not make a difference to the answer in this question.<br>Therefore other possible correct formulae include Area = length x width \u00f7 100 and Area = (length x width) \u00f7 100.\\\"\\n},\\n{\\n\\\"qn\\\":\\\"Which formula can be applied to their measurements to correctly give the perimeter of the handball court in centimetres?\\\",\\n\\\"quant\\\":\\\"perimeter\\\",\\n\\\"ans\\\":\\\"Perimeter = 2 x (length x 100 + width)\\\",\\n\\\"dis1\\\":\\\"Perimeter = 2 x length \u00f7 100 + 2 x width\\\",\\n\\\"dis2\\\":\\\"Perimeter = 2 x (length \u00f7 100 + width)\\\",\\n\\\"dis3\\\":\\\"Perimeter = 2 x width x (length x 100)\\\",\\n\\\"adv\\\":\\\"The answer is to be given in centimetres, so the length needs to be converted from metres to centimetres by multiplying by 100.<br>The perimeter of any rectangle is found by adding the length and the width, then doubling this result.<br>So Perimeter = 2 x (length x 100 + width) is the correct formula.<br>The brackets are necessary to ensure that the length and width are added together before the doubling takes place.<br>Note that the length and width could be each doubled first, then added together.<br>Therefore other possible correct formulae are Perimeter = 2 x length x 100 + 2 x width and Perimeter = (2 x length x 100) + (2 x width).\\\"\\n}\\n]\"))", "description": "", "templateType": "json", "can_override": false}, "scenario": {"name": "scenario", 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"Ungrouped variables", "definition": "scenario[\"dis3\"]", "description": "", "templateType": "anything", "can_override": false}, "people": {"name": "people", "group": "Ungrouped variables", "definition": "random_people(2)", "description": "", "templateType": "anything", "can_override": false}, "person1": {"name": "person1", "group": "Ungrouped variables", "definition": "people[1]", "description": "", "templateType": "anything", "can_override": false}, "name1": {"name": "name1", "group": "Ungrouped variables", "definition": "person1[\"name\"]", "description": "", "templateType": "anything", "can_override": false}, "name2": {"name": "name2", "group": "Ungrouped variables", "definition": "person2[\"name\"]", "description": "", "templateType": "anything", "can_override": false}, "person2": {"name": "person2", "group": "Ungrouped variables", "definition": "people[0]", "description": "", "templateType": "anything", "can_override": false}, "quantity": {"name": "quantity", "group": "Ungrouped 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Give your answer to 2 decimal places.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "{area}+0.1", "minValue": "{area}-0.1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>The diagram below shows a circle.</p>\n<p>{image('resources/question-resources/'+circle)}</p>\n<p>The radius r = {r} cm</p>\n<p>Take $\\pi$ to be 3.14.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"circle": {"definition": "'circle_EeFVDNQ.png'", "templateType": "anything", "group": "Ungrouped variables", "name": "circle", "description": ""}, "r": {"definition": "random(1..25)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "circumference": {"definition": "2*3.14*r", "templateType": "anything", "group": 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Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "<p>Area and Perimeter of Rectangles</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Give answers correct to <strong>2 decimal places</strong></p>", "advice": "<p><strong>(a)</strong></p>\n<p>Area&nbsp;= width $\\times$ length&nbsp;</p>\n<p>width = $\\frac{\\text{area}}{\\text{length} } =\\frac{\\var{area1}}{\\var{h1} } = \\var{w1}$</p>\n<p>Perimeter = $2 \\times$ width&nbsp;$+ 2 \\times$ length$ =2 \\times \\var{w1} + 2 \\times&nbsp;\\var{h1} = \\var{p1}$m</p>\n<p><strong>(b)</strong></p>\n<p>Perimeter = $2 \\times$ width&nbsp;$+ 2 \\times$ length</p>\n<p>Therefore;</p>\n<p>width = $\\frac{\\text{perimeter} -&nbsp;2 \\times \\text{length}}{2}$</p>\n<p>width = $\\frac{(\\var{p} -&nbsp;2 \\times \\var{h})}{2} = \\var{w}$m</p>\n<p></p>\n<p>Formula for area&nbsp;of rectangle:</p>\n<p>Area&nbsp;= width 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