// Numbas version: exam_results_page_options {"name": "Maths Support: Numerical reasoning - ratio and percentage", "navigation": {"onleave": {"action": "none", "message": "

You haven't submitted an answer to this question.

"}, "reverse": false, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "

Based on section 3 of the maths-aid/mathcentre workbook on numerical reasoning.

", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Numerical Reasoning - percentage enlargement", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["dir1", "dir2", "final", "prop1", "prop2", "prop2rel", "verbed1", "verbed2"], "tags": ["maths-aid", "numerical reasoning", "percentage", "Proportion", "proportion", "ratio"], "advice": "

The picture has been {verbed1} to {prop1}% or $\\simplify{{prop1}/100} \\left(=\\frac{\\var{prop1}}{100}\\right)$ of its original size.

\n

So to find the size of the first copy we multiply the original size by $\\simplify{{prop1}/100}$.

\n

If we then {verbed2} the new size by {prop2}%, the final copy would be {prop2rel}% of the size of the first copy, i.e. $\\simplify{{prop2rel}/100}$ of the size of the first copy.

\n

To get the size of the final copy as a proportion of the size of the original copy, we multiply $\\simplify{{prop1}/100}$ by $\\simplify{{prop2rel}/100}$ to get

\n

\$\\simplify{{prop1}/100} \\times \\simplify{{prop2rel}/100} = \\simplify{{prop1*prop2rel}/10000}\$

\n

Now to express this as a percentage we multiply by 100 and we obtain:

\n

\$\\simplify{{prop1*prop2rel}/10000} \\times 100 = \\var{final}\\%.\$

\n

So the final copy is {final}% of the size of the original picture.

", "rulesets": {}, "parts": [{"prompt": "

What percentage of the size of the original picture was the final copy?

\n

[] %

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "final", "minValue": "final", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

A picture on a page was {verbed1} on a copier to {prop1}% of its original size, and this copy was then {verbed2} by {prop2}%.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"dir2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "dir2", "description": ""}, "dir1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "dir1", "description": ""}, "prop2rel": {"definition": "100+dir2*prop2", "templateType": "anything", "group": "Ungrouped variables", "name": "prop2rel", "description": ""}, "prop1": {"definition": "100+dir1*random(10..40#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "prop1", "description": ""}, "prop2": {"definition": "random(10..40#10)", "templateType": "randrange", "group": "Ungrouped variables", "name": "prop2", "description": ""}, "verbed2": {"definition": "if(dir2=1,'increased','reduced')", "templateType": "anything", "group": "Ungrouped variables", "name": "verbed2", "description": ""}, "verbed1": {"definition": "if(dir1=1,'increased','reduced')", "templateType": "anything", "group": "Ungrouped variables", "name": "verbed1", "description": ""}, "final": {"definition": "(prop1/100)*(100+dir2*prop2)", "templateType": "anything", "group": "Ungrouped variables", "name": "final", "description": ""}}, "metadata": {"notes": "", "description": "

Scale a page to some percentage of its original size, then increase/decrease by another percentage. Find the size of the final copy as a percentage of the original.

\n

Based on question 2 from section 3 of the Maths-Aid workbook on numerical reasoning.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numerical Reasoning - percentage increase in price", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "tags": ["maths-aid", "money", "numerical reasoning", "percentage", "profit"], "advice": "

Think of the cost to produce as being 100%. The price is {percent}% greater, so is {percent+100}% of the cost, i.e. $\\frac{\\var{percent+100}}{100}$ of the cost.

\n

Hence the cost is $\\frac{100}{\\var{percent+100}} = \\simplify{100/{percent+100}}$ of the price = $\\simplify{100/{percent+100}} \\times £\\var{dpformat(sell,2)} = £\\var{produce}$.

\n

The profit = selling price - cost = £{dpformat(sell,2)} - £{produce} = £{dpformat(sell-produce,2)}.

", "rulesets": {}, "parts": [{"prompt": "

How much did it cost to produce the {thing} and what was the profit?

\n

Cost to produce: £ []

\n

Profit: £ []

", "gaps": [{"minvalue": "produce", "type": "numberentry", "maxvalue": "produce", "marks": 1.0, "showPrecisionHint": false}, {"minvalue": "sell-produce", "type": "numberentry", "maxvalue": "sell-produce", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "

The selling price of {thing} is £{dpformat(sell,2)}.

\n

This price was {percent}% greater than the cost to produce the {thing}.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"sell": {"definition": "produce*(1+percent/100)", "name": "sell"}, "thing": {"definition": "random(['a box of chocolates','box'],['an action figure','toy'],['a scarf','scarf'])", "name": "thing"}, "produce": {"definition": "random(3..12 except 10)", "name": "produce"}, "percent": {"definition": "random(10..60#5)", "name": "percent"}}, "metadata": {"notes": "", "description": "

Given the selling price of an item both as a cash amount and as a percentage of the cost of production, find the cost of production and the profit.

\n

Based on question 1 from section 3 of the Maths-Aid workbook on numerical reasoning.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numerical reasoning - percentages and profit", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {"commanumber": {"definition": "var parts=n.toString().split(\".\");\n return parts.replace(/\\B(?=(\\d{3})+(?!\\d))/g, \",\") + (parts ? \".\" + parts : \"\");", "type": "string", "language": "javascript", "parameters": [["n", "number"]]}, "lcommanumber": {"definition": "var parts=n.toString().split(\".\");\n return parts.replace(/\\B(?=(\\d{3})+(?!\\d))/g, \",\\\\!\") + (parts ? \".\" + parts : \"\");", "type": "string", "language": "javascript", "parameters": [["n", "number"]]}}, "tags": ["maths-aid", "money", "numerical reasoning", "profit"], "advice": "

First, work out the profit made on the original product.

\n

The original production cost was $\\var{produce}$ {pence} per unit. The profit per unit was $\\var{sell}-\\var{produce} = \\var{sell-produce}$ {pence} per unit.

\n

{commanumber(units1)} units of the original product were sold per month, so the total profit per month was

\n

\$\\var{latex(lcommanumber(units1))} \\times \\var{sell-produce} \\var{p} = \\var{latex(lcommanumber(profit1))} \\var{p} = \\var{latex(texpounds)} \\var{latex(lcommanumber(profit1/100))}. \$

\n

Now work out the cost of producing the new product. The new product costs $\\var{percent}\\%$ more. $\\var{percent}\\%$ of $\\var{produce} = \\simplify{{percent}/100}$ of $\\var{produce} = \\var{produce2-produce}\\var{p}$.

\n

So the new production cost is $\\var{produce} + \\var{produce2-produce} = \\var{produce2} \\var{p}$. The profit per unit is now $\\var{sell} - \\var{produce2} = \\var{sell-produce2} \\var{p}$.

\n

{commanumber(units2)} units of the new product were sold per month, so the total profit per month is now

\n

\$\\var{latex(lcommanumber(units2))} \\times \\var{sell-produce2} \\var{p} = \\var{latex(lcommanumber(profit2))} \\var{p} = \\var{latex(texpounds)} \\var{latex(lcommanumber(profit2/100))}. \$

\n

So the added profit is $\\var{latex(texpounds)}\\var{latex(lcommanumber(profit2/100))} - \\var{latex(texpounds)}\\var{latex(lcommanumber(profit1/100))} = \\var{latex(texpounds)}\\var{latex(lcommanumber(extraprofit))}.$

", "rulesets": {}, "parts": [{"prompt": "

If the manufacturer's selling price in each instance was {sell} {pence} per unit, what was the manufacturer's added profit per month with the newer product?

\n

{pounds} []

", "gaps": [{"minvalue": "extraprofit", "type": "numberentry", "maxvalue": "extraprofit", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "

A product costing {produce} {pence} per unit to produce had been selling at the average rate of {commanumber(units1)} units per month.

\n

After the product was improved, sales increased to an average of {commanumber(units2)} units per month. However, the new product cost {percent} percent more to produce.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"sell": {"definition": "produce+random(20..40#5)", "name": "sell"}, "units1": {"definition": "random(1..15)*mult", "name": "units1"}, "units2": {"definition": "ceil(profit1/(sell-produce2)/mult)*mult+random(1..8)*mult", "name": "units2"}, "produce2": {"definition": "produce*diff", "name": "produce2"}, "pounds": {"definition": "currency", "name": "pounds"}, "texpounds": {"definition": "latex(if(pounds='$','\\\\$',pounds))", "name": "texpounds"}, "percent": {"definition": "(diff-1)*100", "name": "percent"}, "pence": {"definition": "currency", "name": "pence"}, "p": {"definition": "currency", "name": "p"}, "produce": {"definition": "random(40..95#5)", "name": "produce"}, "extraprofit": {"definition": "(profit2-profit1)/100", "name": "extraprofit"}, "diff": {"definition": "random(1.05..floor(20*sell/produce)/20#0.05)", "name": "diff"}, "currency": {"definition": "random(['$','cents','\u00a2'],['\u00a3','pence','p'],['\u20ac','cents','c'])", "name": "currency"}, "profit1": {"definition": "units1*(sell-produce)", "name": "profit1"}, "mult": {"definition": "10^random(3,4,5)", "name": "mult"}, "profit2": {"definition": "ceil(units2*(sell-produce2))", "name": "profit2"}}, "metadata": {"notes": "", "description": " Given cost of production and price of sale of a product; a percentage increase in cost of production; and unit sales before and after; work out the extra profit. \n Based on question 6 from section 3 of the maths-aid workbook on numerical reasoning. ", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numerical reasoning - percentages of subsets", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "tags": ["numerical reasoning", "percentages", "ratio"], "advice": " Calculate separately the percentage of employees who are female and working on the project, and the percentage who are male and working on the project, and add them together. \n #### 1. Females \n {females}% of the department is female and {femaleproject}% of females are working on the project, hence the proportion of workers who are female and working on the project is {femaleproject}% of {females}% of the workers. \n In terms of fractions this is \$\\frac{\\var{femaleproject}}{100} \\times \\frac{\\var{females}}{100} = \\frac{\\var{femaleproject*females}}{10000} = \\frac{\\var{depfemaleproject}}{100}\$ of the workers, ie {depfemaleproject}%. \n So {depfemaleproject}% of the departmental staff are working on the project and female. \n #### 2. Males \n \n {males}% of the department is male and {maleproject}% of males are working on the project, hence the proportion of workers who are male and working on the project is {maleproject}% of {males}% of the workers. \n In terms of fractions this is \$\\frac{\\var{maleproject}}{100} \\times \\frac{\\var{males}}{100} = \\frac{\\var{maleproject*males}}{10000} = \\frac{\\var{depmaleproject}}{100}\$ of the workers, ie {depmaleproject}%. \n So {depmaleproject}% of the departmental staff are working on the project and male. \n So the total percentage of departmental staff working on the project is$\\var{depfemaleproject}\\% + \\var{depmaleproject}\\% = \\var{project}\\%$. \n ", "rulesets": {}, "parts": [{"prompt": " What percentage of the department is working on the project? \n [] % ", "gaps": [{"minvalue": "{project}", "type": "numberentry", "maxvalue": "{project}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": " In a certain department, {femaleproject}% of the females and {maleproject}% of the males are working on a project. {if(describefemales,females,males)}% of the department is {if(describefemales,'female','male')}. ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"females": {"definition": "//percentage of all employees who are female\n random(20..80#5)", "name": "females"}, "describefemales": {"definition": "//Give the percentage of females in the statement, or give the males?\n random(true,false)", "name": "describefemales"}, "femaleproject": {"definition": "//percentage of females working on the project\n random(10..90#10)", "name": "femaleproject"}, "males": {"definition": "//percentage of all employees who are male\n 100-females", "name": "males"}, "depmaleproject": {"definition": "//proportion of employees who are male and working on the project\n maleproject*males/100", "name": "depmaleproject"}, "project": {"definition": "//total percentage of the department working on the project\n (females*femaleproject+males*maleproject)/100", "name": "project"}, "depfemaleproject": {"definition": "//proportion of employees who are female and working on the project\n femaleproject*females/100", "name": "depfemaleproject"}, "maleproject": {"definition": "//percentage of males working on the project\n random(10..90#10)", "name": "maleproject"}}, "metadata": {"notes": "", "description": " Given percentages of males and females working on a project, and the percentage of the total staff who are male (or female), find the percentage of all staff working on the project. \n Based on question 3 from section 3 of the maths-aid workbook on numerical reasoning. ", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numerical reasoning - prices in ratios", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "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"]]}}, "tags": ["constraints", "numerical reasoning", "ratio", "simultaneous equations"], "advice": " Here are two solutions. The first uses the idea of shares and the second uses algebra. \n #### Solution 1 (shares) \n We are given that the cost of \$\\var{ratio[dirs]/gcd(ratio[dirs],ratio[dirs])} \\times \\var{ops[dirs]} = \\var{ratio[dirs]/gcd(ratio[dirs],ratio[dirs])} \\times \\var{ops[dirs]} \$ and \$\\var{ratio[dirs]/gcd(ratio[dirs],ratio[dirs])} \\times \\var{ops[dirs]} = \\var{ratio[dirs]/gcd(ratio[dirs],ratio[dirs])} \\times \\var{ops[dirs]}. \$ \n We can represent the ratios of the costs for {ops[dirs]}, {ops[dirs]} and {ops[dirs]} by giving cost shares to each of the repairs in the ratios {ratio[dirs]}:{ratio[dirs]}:{ratio[dirs]}. \n That is, give {ops[dirs]} {ratio[dirs]} {pluralise(ratio[dirs],'share','shares')}, {ops[dirs]} {ratio[dirs]} {pluralise(ratio[dirs],'share','shares')} and {ops[dirs]} {ratio[dirs]} {pluralise(ratio[dirs],'share','shares')}. Then the relative costs are preserved. \n Hence there are$\\var{ratio[dirs]}+\\var{ratio[dirs]}+\\var{ratio[dirs]} = \\var{ratiototal}$shares to add up to £{total}. \n So each share is worth$\\var{total} \\div \\var{ratiototal} = £\\var{factor}$and {ops[wanted]} gets {ratio[wanted]} {pluralise(ratio[wanted],'share','shares')} i.e. costs £{prices[wanted]}. \n #### Solution 2 (algebra) \n Let$\\var{letters}$= {ops},$\\var{letters}$= {ops},$\\var{letters}$= {ops}. \n We are given that$\\var{letters[dirs]} = \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\var{letters[dirs]}$, and$\\var{letters[dirs]} = \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\var{letters[dirs]}$. \n Rearrange the second equation to give$\\var{letters[dirs]}$in terms of$\\var{letters[dirs]}$: \n \$\\var{letters[dirs]} = \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\var{letters[dirs]} \$ \n So \$\\begin{eqnarray} \\textrm{total cost of repair work} &=& \\var{letters[dirs]} + \\var{letters[dirs]} + \\var{letters[dirs]} \\\\ &=& \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\var{letters[dirs]} + \\var{letters[dirs]} + \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\var{letters[dirs]} \\\\ &=& \\left( \\simplify{{ratio[dirs]}/{ratio[dirs]}} + 1 + \\simplify{{ratio[dirs]}/{ratio[dirs]}} \\right) \\var{letters[dirs]} \\\\ &=& \\simplify{{ratiototal}/{ratio[dirs]}} \\var{letters[dirs]}. \\end{eqnarray} \$ \n Hence$\\simplify{{ratiototal}/{ratio[dirs]}} \\var{letters[dirs]} = £\\var{total}$gives us$\\var{letters[dirs]} = \\simplify{{ratio[dirs]}/{ratiototal}} \\times £\\var{total} = £\\var{prices[dirs]}$. \n So the cost of {ops[wanted]} was £{prices[wanted]}. ", "rulesets": {}, "parts": [{"prompt": " What did {ops[wanted]} cost? \n £ [] ", "gaps": [{"minvalue": "{prices[wanted]}", "type": "numberentry", "maxvalue": "{prices[wanted]}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": " The total cost for three items of work on a {car} was £{total}. \n These items were: {ops}, {ops} and {ops}. \n {capitalise(ops[dir1])} costs {describefraction(ratio[dir1],ratio[dir1])} {ops[dir1]}. \n {capitalise(ops[dir2])} costs {describefraction(ratio[dir2],ratio[dir2])} {ops[dir2]}. ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"dirs": {"definition": "shuffle([0,1,2])", "name": "dirs"}, "dir2": {"definition": "[dirs,dirs]", "name": "dir2"}, "dir1": {"definition": "[dirs,dirs]", "name": "dir1"}, "wanted": {"definition": "dirs", "name": "wanted"}, "letters": {"definition": "map(allops[j],j,0..2)", "name": "letters"}, "r1": {"definition": "random(possibleratios)", "name": "r1"}, "r2": {"definition": "random(possibleratios except r1)", "name": "r2"}, "ops": {"definition": "map(allops[j],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],ratio[dirs])/ratio[dirs]", "name": "f2"}, "f1": {"definition": "lcm(ratio[dirs],ratio[dirs])/ratio[dirs]", "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": " 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. \n Based on question 4 from section 3 of the Maths-Aid workbook on numerical reasoning. ", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numerical reasoning - ratio recipe", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {"describesol": {"definition": "if(ratios[j]=1,\"$U$is not more than$\"+u[j]+\"$.\",\"$\"+ratios[j]+\"$is not more than$\"+u[j]+\"$, i.e.$U$is not more than$\"+(u[j]/ratios[j])+\"$.\")", "type": "string", "language": "jme", "parameters": [["j", "number"]]}}, "tags": ["chain rule", "proportion", "ratio"], "type": "question", "advice": " The proportions {ratios}:{ratios}:{ratios} have to be preserved. \n So if we use$\\simplify{{ratios}*U}${units} of$x$then we must use$\\simplify{{ratios}*U}${units} of$y$and$\\simplify{{ratios}*U}${units} of$z$, to get$\\var{ratiototal}U${units} of the preparation. \n We would like$U$to be as big as possible. \n As we have$\\var{u}${units} of$x$, {describesol(0)} \n As we have$\\var{u}${units} of$y$, {describesol(1)} \n As we have$\\var{u}${units} of$z$, {describesol(2)} \n So the maximum value of$U$is$\\var{lots}$and we can make$\\var{lots} \\times \\var{ratiototal} = \\var{amount}${units} of the preparation. ", "rulesets": {}, "parts": [{"prompt": " How many {units} of the preparation can be made from a stock of materials consisting of {u} {units} of$x$, {u} {units} of$y$, and {u} {units} of$z$? \n [] {units} ", "marks": 0, "gaps": [{"marks": 1, "maxValue": "amount", "minValue": "amount", "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": " A certain preparation consists of liquids$x$,$y$and$z\$ in the proportion {ratios}:{ratios}:{ratios}.

", "variable_groups": [], "progress": "ready", "preamble": {"css": "", "js": ""}, "variables": {"rv": {"definition": "vector(ratios)", "templateType": "anything", "group": "Ungrouped variables", "name": "rv", "description": ""}, "rawratios": {"definition": "shuffle([random(1..7 except 3),random(1..7 except 3),3])", "templateType": "anything", "group": "Ungrouped variables", "name": "rawratios", "description": ""}, "lots": {"definition": "floor(min(map(u[j]/ratios[j],j,0..2)\t))", "templateType": "anything", "group": "Ungrouped variables", "name": "lots", "description": ""}, "uv": {"definition": "vector(u)", "templateType": "anything", "group": "Ungrouped variables", "name": "uv", "description": ""}, "rgcd": {"definition": "gcd(gcd(rawratios,rawratios),rawratios)", "templateType": "anything", "group": "Ungrouped variables", "name": "rgcd", "description": ""}, "amount": {"definition": "lots*ratiototal", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "u": {"definition": "//amount of each liquid\n map(random(3..10)*ratios[j],j,0..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "ratios": {"definition": "map(rawratios[j]/rgcd,j,0..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ratios", "description": ""}, "units": {"definition": "random('litres','gallons','millilitres')", "templateType": "anything", "group": "Ungrouped variables", "name": "units", "description": ""}, "ratiototal": {"definition": "sum(ratios)", "templateType": "anything", "group": "Ungrouped variables", "name": "ratiototal", "description": ""}}, "metadata": {"notes": "", "description": "

Given ratio of ingredients in a preparation, and amounts of each ingredient, work out how much of the preparation you can make.

\n

Based on question 5 from section 3 of the maths-aid workbook on numerical reasoning.

", "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/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}