// Numbas version: exam_results_page_options
{"name": "Numerical reasoning - prices in ratios", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"describefraction": {"definition": "//put fraction into words\n      \n      numbers = ['zero','one','two','three','four','five','six','seven','eight','nine','ten'];\n      denominators = ['','','half','third','quarter','fifth','sixth','seventh','eighth','ninth','tenth'];\n      \n      var gcd = Numbas.math.gcf(n,d);\n      n /= gcd;\n      d /= gcd;\n      \n      if(n%d==0) {\n        var t = n/d;\n        switch(t) {\n      case 1:\n        return 'the same as';\n      case 2:\n        return 'twice as much as';\n      default:\n        return numbers[t]+' times as much as';\n        }\n      }\n      else if(n>d) {\n        var t = (n-(n%d))/d;\n        var m = n%d;\n        if(m==1)\n        return numbers[t]+'-and-a-'+denominators[d]+' times as much as';\n        else\n      return numbers[t]+'-and-'+numbers[m]+'-'+denominators[d]+(m>1?'s':'')+' times as much as';\n      }\n      else if(d==2) {\n        return 'half as much as';\n      }\n      else {\n        return numbers[n]+'-'+denominators[d]+(n>1?'s':'')+' as much as';\n      }", "type": "string", "language": "javascript", "parameters": [["n", "number"], ["d", "number"]]}, "pluralise": {"definition": "return Numbas.util.pluralise(n,singular,plural);", "type": "string", "language": "javascript", "parameters": [["n", "number"], ["singular", "string"], ["plural", "string"]]}, "capitalise": {"definition": "return Numbas.util.capitalise(s);", "type": "string", "language": "javascript", "parameters": [["s", "string"]]}}, "name": "Numerical reasoning - prices in ratios", "tags": ["constraints", "numerical reasoning", "ratio", "simultaneous equations"], "advice": "<p><em>Here are two solutions. The first uses the idea of shares and the second uses algebra.</em></p>\n    <h4>Solution 1 (shares)<em><br/></em></h4>\n    <p>We are given that the cost of \\[ \\var{ratio[dirs[1]]/gcd(ratio[dirs[0]],ratio[dirs[1]])} \\times \\var{ops[dirs[0]]} = \\var{ratio[dirs[0]]/gcd(ratio[dirs[0]],ratio[dirs[1]])} \\times \\var{ops[dirs[1]]} \\] and \\[ \\var{ratio[dirs[2]]/gcd(ratio[dirs[1]],ratio[dirs[2]])} \\times \\var{ops[dirs[1]]} = \\var{ratio[dirs[1]]/gcd(ratio[dirs[1]],ratio[dirs[2]])} \\times \\var{ops[dirs[2]]}. \\]</p>\n    <p>We can represent the ratios of the costs for <strong>{ops[dirs[0]]}</strong>, <strong>{ops[dirs[1]]}</strong> and <strong>{ops[dirs[2]]}</strong> by giving cost shares to each of the repairs in the ratios <strong>{ratio[dirs[0]]}:{ratio[dirs[1]]}:{ratio[dirs[2]]}</strong>.</p>\n    <p>That is, give {ops[dirs[0]]} {ratio[dirs[0]]} {pluralise(ratio[dirs[0]],'share','shares')}, {ops[dirs[1]]} {ratio[dirs[1]]}&nbsp;{pluralise(ratio[dirs[1]],'share','shares')}&nbsp;and {ops[dirs[2]]} {ratio[dirs[2]]}&nbsp;{pluralise(ratio[dirs[2]],'share','shares')}.&nbsp;Then the relative costs are preserved.</p>\n    <p>Hence there are $\\var{ratio[dirs[0]]}+\\var{ratio[dirs[1]]}+\\var{ratio[dirs[2]]} = \\var{ratiototal}$ shares to add up to &pound;{total}.</p>\n    <p>So each share is worth $\\var{total} \\div \\var{ratiototal} = &pound;\\var{factor}$ and {ops[wanted]} gets {ratio[wanted]} {pluralise(ratio[wanted],'share','shares')} i.e. costs <strong>&pound;{prices[wanted]}</strong>.</p>\n    <h4>Solution 2 (algebra)</h4>\n    <p>Let $\\var{letters[0]}$ =&nbsp;{ops[0]}, $\\var{letters[1]}$ = {ops[1]}, $\\var{letters[2]}$ = {ops[2]}.</p>\n    <p>We are given that $\\var{letters[dirs[0]]} = \\simplify{{ratio[dirs[0]]}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]}$, and $\\var{letters[dirs[1]]} = \\simplify{{ratio[dirs[1]]}/{ratio[dirs[2]]}} \\var{letters[dirs[2]]}$.</p>\n    <p>Rearrange the second equation to give $\\var{letters[dirs[2]]}$ in terms of $\\var{letters[dirs[1]]}$:</p>\n    <p>\\[ \\var{letters[dirs[2]]} = \\simplify{{ratio[dirs[2]]}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]} \\]</p>\n    <p>So \\[ \\begin{eqnarray} \\textrm{total cost of repair work} &amp;=&amp; \\var{letters[dirs[0]]} + \\var{letters[dirs[1]]} + \\var{letters[dirs[2]]} \\\\ &amp;=&amp; \\simplify{{ratio[dirs[0]]}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]} + \\var{letters[dirs[1]]} + \\simplify{{ratio[dirs[2]]}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]} \\\\ &amp;=&amp; \\left( \\simplify{{ratio[dirs[0]]}/{ratio[dirs[1]]}} + 1 + \\simplify{{ratio[dirs[2]]}/{ratio[dirs[1]]}} \\right) \\var{letters[dirs[1]]} \\\\ &amp;=&amp; \\simplify{{ratiototal}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]}. \\end{eqnarray} \\]</p>\n    <p>Hence $\\simplify{{ratiototal}/{ratio[dirs[1]]}} \\var{letters[dirs[1]]} = &pound;\\var{total}$ gives us $\\var{letters[dirs[1]]} = \\simplify{{ratio[dirs[1]]}/{ratiototal}} \\times &pound;\\var{total} = &pound;\\var{prices[dirs[1]]}$.</p>\n    <p>So the cost of {ops[wanted]} was <strong>&pound;{prices[wanted]}</strong>.</p>", "rulesets": {}, "parts": [{"prompt": "<p>What did {ops[wanted]} cost?</p>\n        <p>&pound; [[0]]</p>", "gaps": [{"minvalue": "{prices[wanted]}", "type": "numberentry", "maxvalue": "{prices[wanted]}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": ["stats"], "statement": "<p>The total cost for three items of work on a {car} was <strong>&pound;{total}</strong>.</p>\n  <p>These items were: {ops[0]}, {ops[1]} and {ops[2]}.</p>\n  <p>{capitalise(ops[dir1[0]])} costs {describefraction(ratio[dir1[0]],ratio[dir1[1]])} {ops[dir1[1]]}.</p>\n  <p>{capitalise(ops[dir2[0]])} costs {describefraction(ratio[dir2[0]],ratio[dir2[1]])} {ops[dir2[1]]}.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"dirs": {"definition": "shuffle([0,1,2])", "name": "dirs"}, "dir2": {"definition": "[dirs[1],dirs[2]]", "name": "dir2"}, "dir1": {"definition": "[dirs[0],dirs[1]]", "name": "dir1"}, "wanted": {"definition": "dirs[1]", "name": "wanted"}, "letters": {"definition": "map(allops[j][1],j,0..2)", "name": "letters"}, "r1": {"definition": "random(possibleratios)", "name": "r1"}, "r2": {"definition": "random(possibleratios except r1)", "name": "r2"}, "ops": {"definition": "map(allops[j][0],j,0..2)", "name": "ops"}, "allops": {"definition": "shuffle([['overhauling the carburettor','C'],['replacing the brake pads','B'],['refilling the air-con','A'],['replacing the gearbox','G'],['balancing the wheels','W']])[0..3]", "name": "allops"}, "r3": {"definition": "random(possibleratios except [r1,r2])", "name": "r3"}, "f2": {"definition": "lcm(ratio[dirs[0]],ratio[dirs[2]])/ratio[dirs[2]]", "name": "f2"}, "f1": {"definition": "lcm(ratio[dirs[0]],ratio[dirs[2]])/ratio[dirs[0]]", "name": "f1"}, "total": {"definition": "ratiototal*factor", "name": "total"}, "possibleratios": {"definition": "[1,2,3,4,5,6,7,8]", "name": "possibleratios"}, "prices": {"definition": "map(ratio[j]*factor,j,[0,1,2])", "name": "prices"}, "gcdr": {"definition": "//gcd of r1,r2,r3\n    gcd(gcd(r1,r2),r3)", "name": "gcdr"}, "ratiototal": {"definition": "sum(ratio)", "name": "ratiototal"}, "factor": {"definition": "random(12..20)", "name": "factor"}, "ratio": {"definition": "[r1/gcdr,r2/gcdr,r3/gcdr]", "name": "ratio"}}, "metadata": {"notes": "", "description": "<p>Three items of work done on a car. Given total price, and a couple of ratios of prices between pairs of items, work out the cost of one of the items.</p>\n            <p>Based on question 4 from section 3 of the Maths-Aid workbook on numerical reasoning.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}