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{"name": "Sample Paper Q3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Sample Paper Q3", "tags": [], "metadata": {"description": "<p>Solving pair of simultaneous (linear) equations</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>In a workshop there are:</p>\n<ul>\n<li>\\( \\var{A1}\\)&nbsp;machines of type A and&nbsp;\\(&nbsp; \\var{B1} \\) machines of type B that can manufacture a total of&nbsp;\\(&nbsp; \\var{T1} \\) components per week.</li>\n</ul>\n<p>In another workshop there are:</p>\n<ul>\n<li>\\( \\var{A2}\\) machines of type A and \\(&nbsp; \\var{B2} \\) machines of type B that can manufacture a total of \\(&nbsp; \\var{T2} \\) components per week.</li>\n</ul>\n<p></p>\n<p>Let \\(a\\) be the number of components that each machine of type A can manufacture per&nbsp;week and \\(b\\) be the number of components that each machine of type B can&nbsp;manufacture per week.</p>", "advice": "<p>a)&nbsp; The equations can be constructed by careful reading of the question:</p>\n<p><strong><span color=\"red\" style=\"color: red;\">\\( \\var{A1}\\)</span> </strong>&nbsp;machines of type A and&nbsp;&nbsp; <strong><span color=\"red\" style=\"color: red;\">\\(\\var{B1} \\)</span></strong> machines of type B that can manufacture a total of&nbsp;<strong><span color=\"red\" style=\"color: red;\">\\(&nbsp; \\var{T1} \\)</span></strong> components per week.</p>\n<p>Gives us:</p>\n<p style=\"text-align: center;\">\\(&nbsp; \\var{A1}a+\\var{B1}b=\\var{T1}&nbsp; \\)</p>\n<p style=\"text-align: left;\">In the same way, the second statement:</p>\n<p><strong><span color=\"red\" style=\"color: red;\">\\( \\var{A2}\\)</span>&nbsp;</strong>&nbsp;machines of type A and&nbsp;&nbsp;&nbsp;<strong><span color=\"red\" style=\"color: red;\">\\(\\var{B2} \\)</span></strong>&nbsp;machines of type B that can manufacture a total of&nbsp;<strong><span color=\"red\" style=\"color: red;\">\\(&nbsp; \\var{T2} \\)</span></strong>&nbsp;components per week.</p>\n<p>Gives us:</p>\n<p style=\"text-align: center;\">\\(&nbsp; \\var{A2}a+\\var{B2}b=\\var{T2}&nbsp; \\)</p>\n<p style=\"text-align: center;\"></p>\n<hr/>\n<p style=\"text-align: left;\">b)&nbsp; The most straightforward methods to solve this system of equations are \"Elimination\" or \"Substitution\".</p>\n<p style=\"text-align: left;\">\"By Elimination\" involves multiplying the equations so that either the co-efficients of \\(a\\) are the same in both equations or the co-efficients of \\(b \\) are the same:</p>\n<p style=\"text-align: left;\">For example, if we multiply the first equation by \\( \\var{A2} \\) and the second by \\( \\var{A1} \\) we get</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\simplify{{A2}*{A1}a+{A2}*{B1}b={A2}*{T1}}&nbsp; \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\simplify{{A1}*{A2}a+{A1}*{B2}b={A1}*{T2}}&nbsp; \\)</p>\n<p style=\"text-align: left;\">If we now subtract, term by term, one equation from the other we eliminate the terms in \\(a\\):</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\simplify{(({A2}*{B1})-({A1}*{B2}))b = ({A2}*{T1})-({A1}*{T2})}&nbsp; \\)</p>\n<p style=\"text-align: left;\">\\( b = \\frac{\\simplify{({A2}*{T1})-({A1}*{T2})}}{\\simplify{(({A2}*{B1})-({A1}*{B2}))}} \\)</p>\n<p style=\"text-align: left;\">\\( b=\\var{b}\\)</p>\n<p style=\"text-align: left;\">We now have to go back to one of the original equations, use our value for \\(b\\) and calculate \\(a\\):</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\var{A1}a+\\var{B1}b=\\var{T1}&nbsp; \\)</p>\n<p style=\"text-align: left;\">becomes:</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\var{A1}a+\\var{B1}(\\var{b})=\\var{T1}&nbsp; \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\var{A1}a+\\simplify{{B1}*{b}}=\\var{T1}&nbsp; \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\var{A1}a=\\var{T1} -\\simplify{{B1}*{b}} \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; \\var{A1}a=\\simplify{{T1} -({B1}*{b})} \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; a=\\frac{\\simplify{{T1} -({B1}*{b})}}{\\var{A1}} \\)</p>\n<p style=\"text-align: left;\">\\(&nbsp; a=\\var{a} \\)</p>\n<p style=\"text-align: left;\">Finally, we state our findings:</p>\n<p style=\"text-align: left;\">Machine type A can produce&nbsp;\\( \\var{a} \\) components per week and&nbsp;Machine type B can produce&nbsp;\\( \\var{b} \\) components.</p>\n<p style=\"text-align: left;\"></p>\n<hr/>\n<p style=\"text-align: left;\">c)&nbsp; The best way to check your values is to substitute those values into BOTH, original equations and ensure that they \"balance\".</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Answers", "definition": "random(35 .. 60#5)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Answers", "definition": "random(15 .. 30#5)", "description": "", "templateType": "randrange"}, "A1": {"name": "A1", "group": "Equation 1", "definition": "random(3..10)", "description": "", "templateType": "anything"}, "B1": {"name": "B1", "group": "Equation 1", "definition": "random(3..10 except A1)", "description": "", "templateType": "anything"}, "T1": {"name": "T1", "group": "Equation 1", "definition": "A1*a + B1*b", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "Equation 2", "definition": "random(3..10 except A1 except B1)", "description": "", "templateType": "anything"}, "B2": {"name": "B2", "group": "Equation 2", "definition": "random(3..10 except A1 except B1 except A2)", "description": "", "templateType": "anything"}, "T2": {"name": "T2", "group": "Equation 2", "definition": "A2*a + B2*b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Equation 1", "variables": ["A1", "B1", "T1"]}, {"name": "Answers", "variables": ["a", "b"]}, {"name": "Equation 2", "variables": ["A2", "B2", "T2"]}], "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 down two simultaneous equations from the given information.</p>\n<p>The first equation uses the information from the first workshop:</p>\n<p style=\"text-align: center;\">[[0]]</p>\n<p style=\"text-align: left;\">The second equation can be writen using the information from the second workshop:</p>\n<p style=\"text-align: center;\">[[1]]</p>", "gaps": [{"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": "{A1}*a+{B1}*b={T1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}, {"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": "{A2}*a+{B2}*b={T2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}]}], "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>Solve these equations to find the number of components each type of machine can&nbsp;manufacture per week.</p>\n<p style=\"text-align: center;\">\\(a=\\) [[0]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(b=\\)&nbsp;[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a}", "maxValue": "{a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2.5", "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, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>What would be a suitable method to check that the values you found are correct?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["Substitute the values into both original equations to ensure that they balance.", "Substitute the values into one of the original equations to ensure that it balances.", "Divide the totals by the values.", "Multiply the values together to see if they equal the totals."], "matrix": ["2", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}