// Numbas version: exam_results_page_options
{"name": "Solution of a word problem using graphical simultaneous equations.", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solution of a word problem using graphical simultaneous equations.", "tags": [], "metadata": {"description": "<p>Students are shown to intersecting lines in the context of a gym visiting program. They are asked to identify the y-intercept, the gradients, the point of intersection and are asked to interpret the graphs in the context of the word problem.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A gym offers casual visits and memberships.</p>\n<p>For a casual visit, you pay a fixed amount each time you visit the gym.</p>\n<p>For a membership, you pay an upfront fee, and then a smaller amount each time you visit the gym.</p>\n<p>The graph&nbsp;the relationship between cost (in dollars) ($y$) and number of visits to the gym ($x$) for the two programs.</p>\n<p></p>\n<p>{geogebra_applet('https://www.geogebra.org/m/uky4vz3z',defs)}</p>", "advice": "<p>The red line represents the gym membership option because the line has a non-zero $y$-intercept which represents the value of the upfront payment.</p>\n<p></p>\n<p>If you intend to visit the gym {visits1}&nbsp;times, <span data-jme-visible=\"(m1*visits1)&lt;(m2*visits1+b2)\">the cheaper option is using casual visits&nbsp;because this line has a lower value at the point where $x=\\var{visits1}$</span>&nbsp;<span data-jme-visible=\"(m1*visits1)&gt;(m2*visits1+b2)\">the cheaper option is using a gym membership because this line has a lower value at the point where $x=\\var{visits1}$</span>&nbsp;<span data-jme-visible=\"(m1*visits1)=(m2*visits1+b2)\">it doesn't matter which option you choose because the cost is the same</span>.</p>\n<p><span data-jme-visible=\"gymwins=1\">The fewest number of visits for which it is cheaper to take out a gym membership is {k+gymwins}. This is the first $x$-axis value for which the gym membership line is below the casual visit line.</span></p>\n<p><span data-jme-visible=\"gymwins=-1\">The most number of visits for which it is cheaper to pay for casual visits is {k+gymwins}. This the last $x$-axis value for which the casual visit line is below the gym membership line.</span></p>\n<p><span data-jme-visible=\"gymwins=0\">{k}&nbsp;visits are required for the cost of a gym membership to equal the cost of the casual visits. This is the point at which the two lines intersect.</span></p>\n<p>The cost of the upfront payment for gym membership is the point at which the gym membership line cuts the $y$-axis, which is at \\${b2}.</p>\n<p><span data-jme-visible=\"visittype=0\">The cost of entry to the gym on a casual visit is the gradient of the casual visit line. It is also the value of the casual visit line when $x$=1 (1 visit). This is equal to {m1}.</span></p>\n<p><span data-jme-visible=\"visittype=1\">The cost of entry to the gym if you own a membership is the gradient of the gym membership line. This is given by vertical rise over horizontal run and is equal to {m2}.</span></p>\n<p>The graph is only meaningful at points where $x$ is a whole number because you cannot have fractional visits to the gym. For example, you cannot visit the gym 6.7 times.</p>", "rulesets": {}, "extensions": ["geogebra"], "variables": {"defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n  ['b1',0],['m1',m1],['b2',b2],['m2',m2],\n  ['xmin',0],['xmax',xmax],['ymin',0],['ymax',ymax]\n  ]", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "random(15,20,25,30)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "m1-b2/k", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(5..10)*10", "description": "", "templateType": "anything"}, "xmax": {"name": "xmax", "group": "Ungrouped variables", "definition": "1.5*k", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(10)", "description": "<p>the x-value of the intersection point</p>", "templateType": "anything"}, "ymax": {"name": "ymax", "group": "Ungrouped variables", "definition": "1.5*m1*k", "description": "", "templateType": "anything"}, "visits1": {"name": "visits1", "group": "Ungrouped variables", "definition": "random(1..xmax)", "description": "", "templateType": "anything"}, "gymwins": {"name": "gymwins", "group": "Ungrouped variables", "definition": "random(-1,0,1)", "description": "", "templateType": "anything"}, "visittype": {"name": "visittype", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["defs", "m1", "m2", "b2", "xmax", "k", "ymax", "visits1", "gymwins", "visittype"], "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>Which line represents the gym membership option?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["red line", "black line"], "matrix": ["1", 0], "distractors": ["", ""]}, {"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>If you intend to visit the gym {visits1}&nbsp;times, which is the cheaper option?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["casual visits", "gym membership", "it doesn't matter because they are the same price"], "matrix": ["if((m1*visits1)<(m2*visits1+b2),1,0)", "if((m1*visits1)>(m2*visits1+b2),1,0)", "if((m1*visits1)=(m2*visits1+b2),1,0)"], "distractors": ["", "", ""]}, {"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><span data-jme-visible=\"gymwins=1\">What is the fewest number of visits for which it is cheaper to take out a gym membership?</span></p>\n<p><span data-jme-visible=\"gymwins=-1\">What is the most number of visits for which it is cheaper to pay for casual visits?</span></p>\n<p><span data-jme-visible=\"gymwins=0\">How many visits are required for the cost of a gym membership to equal the cost of the casual visits?</span></p>", "minValue": "k+gymwins", "maxValue": "k+gymwins", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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>What is the&nbsp;cost of the upfront payment for gym membership?</p>", "minValue": "b2", "maxValue": "b2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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><span data-jme-visible=\"visittype=0\">What is the cost of entry to the gym on a casual visit?</span></p>\n<p><span data-jme-visible=\"visittype=1\">What is the cost of entry to the gym if you own a membership?</span></p>", "minValue": "if(visittype=0,m1,m2)", "maxValue": "if(visittype=0,m1,m2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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>Why is the graph only meaningful for the values where $x$ is a whole number?</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}