// Numbas version: exam_results_page_options {"name": "The order of operations", "metadata": {"description": "

A quick practice set of problems for education students to take in preparation for their numeracy test.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "The order of operations: the four basic operations", "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/"}], "ungrouped_variables": ["a", "b", "c", "ans1", "d", "f", "g", "ans2", "ans3", "list"], "statement": "

Write the following questions down on paper and evaluate them without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "tags": ["order of operations", "precedence"], "variable_groups": [], "rulesets": {}, "parts": [{"customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

$\\var{a}+\\var{b}\\times\\var{c}=$ [[0]]

", "sortAnswers": false, "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

The order of operation dictates that we deal with multiplication (and division) before addition (and subtraction), that is 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{a}+\\var{b}\\times\\var{c}$$=$$\\var{a}+\\var{b*c}$
$=$$\\var{ans1}$
", "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"customMarkingAlgorithm": "", "maxValue": "{ans1}", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "mustBeReduced": false, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "minValue": "{ans1}", "allowFractions": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"]}]}, {"customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

$\\var{d}-\\var{f}\\div\\var{g}=$ [[0]]

", "sortAnswers": false, "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

The order of operation dictates that we deal with multiplication (and division) before addition (and subtraction), that is 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{d}-\\var{f}\\div\\var{g}$$=$$\\var{d}-\\var{a}$
$=$$\\var{ans2}$
", "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"customMarkingAlgorithm": "", "maxValue": "{ans2}", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "mustBeReduced": false, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "minValue": "{ans2}", "allowFractions": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"]}]}, {"customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

$\\var{d}\\times\\var{c}-\\var{b}\\times\\var{a}+\\var{g^2}\\div \\var{g}=$ [[0]]

", "sortAnswers": false, "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "prompt": "

The order of operation dictates that we deal with multiplication and division before addition and subtraction, that is 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{d}\\times\\var{c}-\\var{b}\\times\\var{a}+\\var{g^2}\\div \\var{g}$$=$$\\var{d*c}-\\var{a*b}+\\var{g}$
$=$$\\var{ans3}$
", "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "gaps": [{"customMarkingAlgorithm": "", "maxValue": "{ans3}", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "mustBeReduced": false, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "minValue": "{ans3}", "allowFractions": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"]}]}], "variables": {"ans2": {"description": "", "definition": "d-a", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2"}, "d": {"description": "", "definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d"}, "g": {"description": "", "definition": "list[3]", "templateType": "anything", "group": "Ungrouped variables", "name": "g"}, "c": {"description": "", "definition": "list[3]", "templateType": "anything", "group": "Ungrouped variables", "name": "c"}, "a": {"description": "", "definition": "list[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "a"}, "f": {"description": "", "definition": "a*g", "templateType": "anything", "group": "Ungrouped variables", "name": "f"}, "ans1": {"description": "", "definition": "a+b*c", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1"}, "ans3": {"description": "", "definition": "d*c-a*b+g", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3"}, "b": {"description": "", "definition": "list[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "b"}, "list": {"description": "", "definition": "shuffle(2..12)[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "list"}}, "functions": {}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "advice": "", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}, {"name": "The order of operations: brackets, powers and the four basics", "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/"}], "tags": ["order of operations", "precedence"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Write the following questions down on paper and evaluate them without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "list[0]", "description": "", "templateType": "anything", "can_override": false}, "base": {"name": "base", "group": "Ungrouped variables", "definition": "list[4]", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "list[2]", "description": "", "templateType": "anything", "can_override": false}, "subs": {"name": "subs", "group": "Ungrouped variables", "definition": "sub^2", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "h*(f-g)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "list[3]", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "list[5]", "description": "", "templateType": "anything", "can_override": false}, "tsub": {"name": "tsub", "group": "Ungrouped variables", "definition": "2*sub", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "base+a", "description": "", "templateType": "anything", "can_override": false}, "denom": {"name": "denom", "group": "Ungrouped variables", "definition": "-base+tsub", "description": "", "templateType": "anything", "can_override": false}, "diffs": {"name": "diffs", "group": "Ungrouped variables", "definition": "diff^2", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2,3,4)", "description": "", "templateType": "anything", "can_override": false}, "diff": {"name": "diff", "group": "Ungrouped variables", "definition": "base-sub", "description": "", "templateType": "anything", "can_override": false}, "sub": {"name": "sub", "group": "Ungrouped variables", "definition": "random(1..3 except base/2)", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "a*b^c+d", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "if(c-3>=0,random(1,2,3),random(1..12))", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "list[1]", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "subs-diffs", "description": "", "templateType": "anything", "can_override": false}, "list": {"name": "list", "group": "Ungrouped variables", "definition": "shuffle(2..12)[0..6]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["list", "a", "b", "c", "d", "ans1", "f", "h", "g", "ans2", "base", "sub", "diff", "subs", "diffs", "num", "tsub", "denom", "ans3"], "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": "

$\\var{d}+\\var{a}\\times\\var{b}^\\var{c}=$ [[0]]

", "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": "

The order of operation dictates that we deal with powers before multiplication/division and also deal with multiplication/division before addition/subtraction , that is 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{d}+\\var{a}\\times\\var{b}^\\var{c}$$=$$\\var{d}+\\var{a}\\times\\var{b^c}$
$=$$\\var{d}+\\var{a*b^c}$
$=$$\\var{ans1}$
"}], "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, "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": "

$\\var{h}(\\var{f}-\\var{g})=$ [[0]]

", "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": "

Note: $\\var{h}(\\var{f}-\\var{g})$ means $\\var{h}\\times(\\var{f}-\\var{g})$.

\n

\n

The order of operation dictates that we deal with brackets (grouping symbols) before multiplication, that is 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{h}(\\var{f}-\\var{g})$$=$$\\var{h}(\\var{f-g})$
$=$$\\var{ans2}$
\n

\n

Note however, that the distributive law allows for another method! 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{h}(\\var{f}-\\var{g})$$=$$\\var{h}\\times\\var{f}-\\var{h}\\times\\var{g})$
$=$$\\var{h*f}-\\var{h*g}$
$=$$\\var{ans2}$
\n

"}], "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": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "

$\\displaystyle{\\var{a}+\\frac{\\var{sub}^2-(\\var{base}-\\var{sub})^2}{-\\var{base}+2\\times\\var{sub}}} =$ [[0]]

", "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": "

Note: A fraction $\\frac{a}{b}$ is the same as $(a)\\div (b)$, so we have to evaluate the numerator and denominator before doing the division. We can evaluate the numerator at the same time as we evaluate the denominator.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\var{a}+\\frac{\\var{sub}^2-(\\var{base}-\\var{sub})^2}{-\\var{base}+2\\times\\var{sub}}}$$=$$\\displaystyle{\\var{a}+\\frac{\\var{sub}^2-(\\var{diff})^2}{-\\var{base}+2\\times\\var{sub}}}$(work on the innermost bracketed expression first)
$=$$\\displaystyle{\\var{a}+\\frac{\\var{subs}-\\var{diffs}}{-\\var{base}+2\\times\\var{sub}}}$(doing the powers on the numerator, and multiplication on the denominator)
$=$$\\displaystyle{\\var{a}+\\frac{\\var{num}}{-\\var{base}+\\var{tsub}}}$(doing multiplication on the denominator and addition on the numerator)
$=$$\\displaystyle{\\var{a}+\\frac{\\var{num}}{\\var{denom}}}$(continue working on the denominator)
$=$$\\displaystyle{\\var{a}+\\var{base}}$(do the division, or simplify the fraction)
$=$$\\var{ans3}$(finally do the last addition)
"}], "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, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": false, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "extensions": [], "custom_part_types": [], "resources": []}