Tests Algebra, Graphing Straight Lines, Probability, Statistics
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "30 random students were asked about the number of siblings they have. These are their responses:
\n| $\\var{a[0]}$ | \n$\\var{a[1]}$ | \n$\\var{a[2]}$ | \n$\\var{a[3]}$ | \n$\\var{a[4]}$ | \n$\\var{a[5]}$ | \n$\\var{a[6]}$ | \n$\\var{a[7]}$ | \n$\\var{a[8]}$ | \n$\\var{a[9]}$ | \n
| $\\var{a[10]}$ | \n$\\var{a[11]}$ | \n$\\var{a[12]}$ | \n$\\var{a[13]}$ | \n$\\var{a[14]}$ | \n$\\var{a[15]}$ | \n$\\var{a[16]}$ | \n$\\var{a[17]}$ | \n$\\var{a[18]}$ | \n$\\var{a[19]}$ | \n
| $\\var{a[20]}$ | \n$\\var{a[21]}$ | \n$\\var{a[22]}$ | \n$\\var{a[23]}$ | \n$\\var{a[24]}$ | \n$\\var{a[25]}$ | \n$\\var{a[26]}$ | \n$\\var{a[27]}$ | \n$\\var{a[28]}$ | \n$\\var{a[29]}$ | \n
Organising the data in a frequency table helps to make mistakes less likely when calculating statistics from our data, summarising the responses all in one place with fewer numbers.
\nEach row of the frequency column gives the number of students with the corresponding number of siblings.
\n| Number of siblings | \nFrequency | \n
|---|---|
| $0$ | \n$\\var{freq[0]}$ | \n
| $1$ | \n$\\var{freq[1]}$ | \n
| $2$ | \n$\\var{freq[2]}$ | \n
| $3$ | \n$\\var{freq[3]}$ | \n
| $4$ | \n$\\var{freq[4]}$ | \n
| $5$ | \n$\\var{freq[5]}$ | \n
| $6$ | \n$\\var{freq[6]}$ | \n
| Total | \n$30$ | \n
Always remember to check whether your frequency column adds up to the total (here, it is $30$) to make sure you have not left out any responses.
\nThe mean number of siblings is the total number of siblings, $\\sum x$, divided by the number of students in the sample, $n$.
\n\\begin{align}
\\sum x &= 0 \\times \\var{freq[0]} + 1 \\times \\var{freq[1]} + 2 \\times \\var{freq[2]} + 3 \\times \\var{freq[3]} + 4 \\times \\var{freq[4]} + 5 \\times \\var{freq[5]} + 6 \\times \\var{freq[6]}
\\\\
&= 0 + \\var{1*freq[1]} + \\var{2*freq[2]} + \\var{3*freq[3]} + \\var{4*freq[4]} + \\var{5*freq[5]} + \\var{6*freq[6]} \\\\&= \\var{sum(a)} \\text{.}
\\end{align}
The total number of students $n$ is $30$.
\nTherefore the mean is
\n\\begin{align}
\\bar{x} &= \\frac{\\sum x}{n} \\\\
&= \\frac{\\var{sum(a)}}{30} \\\\
&= \\var{mean} \\text{.}
\\end{align}
Rounding the answer to 2 decimal places, we get $\\var{precround(mean, 2)}$.
\nThe mode is the value with the highest frequency. Here, the mode is $\\var{mode}$ siblings, with frequency $\\var{freq[mode]}$.
\nThe median is the \"middle\" value in the sample, when arranged in numerical order.
\nSince $n = 30$, we have two middle values in this data (15th and 16th place). We can count from the top of the table until we locate rows where these middle values lie, as the numbers in the table are already sorted by order.
\nHere, both $15$th and $16$th value lie in the row $\\var{asa[14]}$.Here, the $15$th value lies in the row $\\var{asa[14]}$ while the $16$th value lies in the row $\\var{asa[15]}$.
\nAs $15$th value $= 16$th value $= \\var{asa[14]}$, the median is $\\var{asa[14]}$.As $15$th value $= \\var{asa[14]}$ and $16$th value $= \\var{asa[15]}$, we need to find their mean.
\n\\[ \\displaystyle \\begin{align} \\frac{\\var{asa[14]} + \\var{asa[15]}}{2} &= \\frac{\\var{asa[14] + asa[15]}}{2} \\\\&= \\var{median} \\text{.} \\end{align}\\]
\nThis is the median for this data.
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\n| Number of siblings | \nFrequency | \n
|---|---|
| $0$ | \n[[0]] | \n
| $1$ | \n[[1]] | \n
| $2$ | \n[[2]] | \n
| $3$ | \n[[3]] | \n
| $4$ | \n[[4]] | \n
| $5$ | \n[[5]] | \n
| $6$ | \n[[6]] | \n
| Total | \n$30$ | \n
Find the mean, mode and median for this data.
\nMean = [[0]]
\nMode = [[1]]
\nMedian = [[2]]
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\nMove the points as follows:
\nA to $(\\var{a1},\\var{a2})$.
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\n", "type": "gapfill"}, {"scripts": {"mark": {"script": "console.log(this.question.points);\nthis.question.points.c.setAttribute({fillColor: this.credit==1 ? 'green' : 'red'});\n", "order": "after"}}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "{c1}", "showFeedbackIcon": true, "minValue": "{c1}", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "scripts": {}, "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "{c2}", "showFeedbackIcon": true, "minValue": "{c2}", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "scripts": {}, "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "C to $(\\var{c1},\\var{c2})$.
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So, for example $(2,5)$ has $x$-coordinate $= 2$ and $y$-coordinate$= 5$.
\nPlot the first coordinate ($x$-coordinate) against the horizontal axis and then plot the second coordinate ($y$-coordinate) against the vertical axis.
\n{correctPoints()}
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", "unitTests": [], "customName": ""}], "advice": "When simplifying expressions, only terms of the same type or like terms can be added together.
\nAlgebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.
\n\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]
\\[
\\begin{align}
\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\
&=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]}
\\end{align}
\\]
\\[
\\begin{align}
\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\
&=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5
\\end{align}
\\]
\\[
\\begin{align}
\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc
&=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\
&=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c
\\end{align}
\\]
\\[
\\begin{align}
\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2
&=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\
&=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab
\\end{align}
\\]
\\[
\\begin{align}
\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y
&=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\
&=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\
&=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y
\\end{align}
\\]
\\[
\\begin{align}
\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2
&=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z
\\end{align}
\\]
\\[
\\begin{align}
\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)
&= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\
&= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\
&= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y}
\\end{align}
\\]
For each expression below, collect like terms and expand brackets.
\nThe * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.
Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.
"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Find the equation of a line through two points - negative gradient", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "variable_groups": [], "rulesets": {}, "functions": {"correctPoints": {"parameters": [], "language": "javascript", "type": "html", "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[math.min(xa-4,-2),math.max(ya+4,2),math.max(xb+4,2),math.min(yb-4,-2)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\n\n\nquestion.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\n//define ans as this \ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}, "plotPoints": {"parameters": [], "language": "javascript", "type": "html", "definition": "\n//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[math.min(xa-4,-2),math.max(ya+4,2),math.max(xb+4,2),math.min(yb-4,-2)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n var ans;\n var nscope = new Numbas.jme.Scope([scope,{variables:{x:new Numbas.jme.types.TNum(0)}}]);\n//this is the beating heart of whatever plots the function,\n//I've changed this from being curve to functiongraph\n var line = board.create('functiongraph',[function(x){\nif(ans) {\n try {\nnscope.variables.x.value = x;\n var val = Numbas.jme.evaluate(ans,nscope).value;\n return val;\n }\n catch(e) {\nreturn 13;\n }\n}\nelse\n return 13;\n },-12,12]\n , {strokeColor:\"blue\",strokeWidth: 4} );\n \nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: false, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\nquestion.lines = {\n l:line, c:correct_line\n}\n\n question.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\nvar expr = question.parts[2].gaps[0].display.studentAnswer();\n\n//define ans as this \ntry {\n ans = Numbas.jme.compile(expr,scope);\n}\ncatch(e) {\n ans = null;\n}\nline.updateCurve();\ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}}, "ungrouped_variables": ["xa", "xb", "ya", "yb", "m", "c", "twos", "twos2"], "metadata": {"description": "Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.
\nThe answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.
\nThis particular example has a negative gradient.
", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "We find the equation of a straight line passing through two points by finding the gradient and the $y$-intercept of the line.
We can find the gradient ($m$) using the points $A$ and $B$, $(x_1,y_1)=(\\var{xa},\\var{ya})$ and $(x_2,y_2)=(\\var{xb},\\var{yb})$ respectively.
The definition of gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$):
\n\\[
\\begin{align}
m &= \\frac{y_2-y_1}{x_2-x_1} \\\\
&= \\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}} \\\\
&= \\frac{\\simplify{{yb}-{ya}}}{\\simplify{{xb}-{xa}}} \\\\
&= \\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}
\\end{align}
\\]
Rearranging the equation $y=mx+c$ and substituting either of the points gives two equations for the $y$-intercept $c$:
\n\\[c = y_1-mx_1 \\quad \\mathrm{or} \\quad c = y_2-mx_2 \\,\\text{,} \\]
\nLet's use point $B$:
\n\\[
\\begin{align}
c &= y_2-mx_2 \\\\
&= \\var{ya}-(\\var{m}\\times\\var{xa}) \\\\
&= \\simplify{{ya-m*xa}}\\text{.}
\\end{align}
\\]
We then check this against point $A$:
\n\\[
\\begin{align}
y_1 &= mx_1 + c \\\\
&= \\simplify[fractionNumbers]{{m}{xb}+{c}} \\\\
&= \\simplify{{m}*{xb}+{c}}\\text{.}
\\end{align}
\\]
Substituting our values for $m$ and $c$ into the equation for a straight line, $y=mx+c$, gives
\n\\[y=\\simplify[all,!noLeadingMinus]{{m} x+ {c}}\\text{.}\\]
\nThis is plotted below:
\n{correctPoints()}
", "statement": "In this question we will identify the equation of the straight line passing through the points $A=(\\var{xa},\\var{ya})$ and $B=(\\var{xb},\\var{yb})$, in the form $y = mx + c$.
\n{plotPoints()}
", "preamble": {"js": "", "css": ""}, "tags": ["gradient", "graphs", "line equation", "negative gradient", "Straight Line", "straight line", "taxonomy", "y-intercept"], "parts": [{"variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "prompt": "Calculate the gradient, $m$, of the line between these two points.
\n$ m=$ [[0]]
\n", "customName": "", "gaps": [{"extendBaseMarkingAlgorithm": true, "useCustomName": false, "correctAnswerStyle": "plain", "maxValue": "m", "customName": "", "showFractionHint": true, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "correctAnswerFraction": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": 1, "allowFractions": false, "minValue": "m", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "useCustomName": false, "adaptiveMarkingPenalty": 0, "unitTests": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 0, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "prompt": "Use this gradient and the points to calculate the $y$-intercept, $c$.
\n$c=$ [[0]]
", "customName": "", "gaps": [{"extendBaseMarkingAlgorithm": true, "useCustomName": false, "correctAnswerStyle": "plain", "maxValue": "c", "customName": "", "showFractionHint": true, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "correctAnswerFraction": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": 1, "allowFractions": false, "minValue": "c", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "useCustomName": false, "adaptiveMarkingPenalty": 0, "unitTests": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 0, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {"mark": {"script": "console.log(this.question.lines.c)\nthis.question.lines.l.setAttribute({strokeColor: this.credit==1 ? 'green' : 'red'});\nthis.question.lines.c.setAttribute({visible: this.credit==1 ? false : true});\n", "order": "after"}}, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "prompt": "Using your values for $m$ and $c$, write down the equation of the straight line which passes through the two points A and B, in the form $y = mx +c$
\n$\\displaystyle y=$ [[0]]
\nUse the graph to plot your answer and check that it goes through these points.
", "customName": "", "gaps": [{"answer": "{m}*x+{c}", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "notallowed": {"message": "You must input your answer in the form y = mx +c where m and c are numbers.
", "showStrings": false, "partialCredit": 0, "strings": ["c", "m"]}, "checkingType": "absdiff", "unitTests": [], "customName": "", "variableReplacements": [], "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "alwaysreplace", "type": "jme", "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 1, "adaptiveMarkingPenalty": 0}], "useCustomName": false, "adaptiveMarkingPenalty": 0, "unitTests": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 0, "showCorrectAnswer": true}], "variables": {"c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "ya-m*xa"}, "yb": {"name": "yb", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "ya-twos2"}, "twos2": {"name": "twos2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4 except 3)"}, "m": {"name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(ya-yb)/(xa-xb)"}, "xb": {"name": "xb", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "xa+twos"}, "twos": {"name": "twos", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4 except 3 )"}, "ya": {"name": "ya", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)"}, "xa": {"name": "xa", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..2 except[0,-2,-4])"}}, "variablesTest": {"maxRuns": 100, "condition": "\n"}, "type": "question"}, {"name": "Find the equation of a line through two points - positive gradient", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "statement": "In this question we will identify the equation of the straight line passing through points $A=(\\var{xa},\\var{ya})$ and $B=(\\var{xb},\\var{yb})$ in the form $y = mx + c$.
\n{plotPoints()}
", "variables": {"m": {"name": "m", "description": "", "definition": "(ya-yb)/(xa-xb)", "templateType": "anything", "group": "Ungrouped variables"}, "yb": {"name": "yb", "description": "", "definition": "ya+random([2,4])", "templateType": "anything", "group": "Ungrouped variables"}, "xa": {"name": "xa", "description": "", "definition": "random(-4..-1)", "templateType": "anything", "group": "Ungrouped variables"}, "ya": {"name": "ya", "description": "", "definition": "random(-4..2)", "templateType": "anything", "group": "Ungrouped variables"}, "xb": {"name": "xb", "description": "", "definition": "xa+random([2,4] except -xa)", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"name": "c", "description": "", "definition": "ya-m*xa", "templateType": "anything", "group": "Ungrouped variables"}}, "metadata": {"description": "Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.
\nThe answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.
\nThis particular example has a positive gradient.
", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["gradient", "graphs", "line equation", "Straight Line", "straight line", "taxonomy", "y-intercept"], "variablesTest": {"maxRuns": 100, "condition": "\n"}, "variable_groups": [], "ungrouped_variables": ["xa", "xb", "ya", "yb", "m", "c"], "advice": "We find the equation of a straight line passing through two points by finding the gradient and the $y$-intercept of the line.
\nWe can find the gradient ($m$) using the points $A = (x_1,y_1)=(\\var{xa},\\var{ya})$ and $B = (x_2,y_2)=(\\var{xb},\\var{yb})$.
\nAs the definition of gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).
The equation for gradient is,
\\begin{align}
m &= \\frac{y_2-y_1}{x_2-x_1} \\\\[0.5em]
&= \\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}} \\\\[0.5em]
&= \\frac{\\simplify[]{{yb}-{ya}}}{\\simplify{{xb}-{xa}}} \\\\[0.5em]
&= \\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}
\\end{align}
Rearranging the equation $y=mx+c$ and substituting either of the points gives
\n\\[c = y_1-mx_1 \\quad \\mathrm{or} \\quad c = y_2-mx_2 \\,\\text{.} \\]
\nWe can then also use this equation with the other point's coordinates to check our answer.
\nLet's use point $A$ first:
\n\\[
\\begin{align}
c &= y_1-mx_1 \\\\
&= \\var{ya}-\\var[fractionnumbers]{m}\\times\\var{xa} \\\\
& = \\simplify[fractionnumbers]{{ya-m*xa}}\\text{.}
\\end{align}
\\]
We then check this against point $B$:
\n\\[
\\begin{align}
y_2 &= mx_2 + c \\\\[0.5em]
&= \\simplify[fractionNumbers]{{m}{xb}+{c}} \\\\[0.5em]
&= \\var[fractionnumbers]{m*xb+c}\\text{.}
\\end{align}
\\]
We can now substitute these values for $m$ and $c$ into $y=mx+c$ to get:
\n\\[y=\\simplify[!noLeadingMinus,fractionNumbers,unitFactor]{{m} x+ {c}}\\text{.}\\]
\nThe green line drawn on the graph represents the above line equation.
\n{correctPoints()}
", "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "prompt": "Calculate the gradient, $m$, of the straight line between these two points.
\n$m=$ [[0]]
\n", "unitTests": [], "scripts": {}, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "gaps": [{"correctAnswerFraction": true, "unitTests": [], "scripts": {}, "adaptiveMarkingPenalty": 0, "correctAnswerStyle": "plain", "minValue": "m", "marks": 1, "mustBeReducedPC": 0, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "allowFractions": true, "mustBeReduced": false, "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "customName": "", "maxValue": "m", "variableReplacementStrategy": "originalfirst"}], "useCustomName": false, "customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "prompt": "Use this gradient and the coordinates of the points to calculate the $y$-intercept, $c$.
\n$c=$ [[0]]
", "unitTests": [], "scripts": {}, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "gaps": [{"correctAnswerFraction": false, "unitTests": [], "scripts": {}, "adaptiveMarkingPenalty": 0, "correctAnswerStyle": "plain", "minValue": "c", "marks": 1, "mustBeReducedPC": 0, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "allowFractions": false, "mustBeReduced": false, "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "customName": "", "maxValue": "c", "variableReplacementStrategy": "originalfirst"}], "useCustomName": false, "customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "prompt": "Give the equation of the straight line through these points in the form $y=mx+c$.
\n$\\displaystyle y=$ [[0]]
\nUse the graph to plot your answer and check that it goes through these points.
", "unitTests": [], "scripts": {"mark": {"order": "after", "script": "this.question.lines.l.setAttribute({strokeColor: this.credit==1 ? 'green' : 'red'});\nthis.question.lines.c.setAttribute({visible: this.credit==1});\n"}}, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "gaps": [{"showPreview": true, "vsetRangePoints": 5, "scripts": {}, "unitTests": [], "checkingType": "absdiff", "vsetRange": [0, 1], "failureRate": 1, "answerSimplification": "fractionNumbers", "useCustomName": false, "answer": "{m}*x+{c}", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "notallowed": {"message": "You must input your answer in the form y = mx +c where m and c are numbers.
", "showStrings": false, "partialCredit": 0, "strings": ["c", "m"]}, "checkVariableNames": true, "checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "type": "jme", "valuegenerators": [{"name": "x", "value": ""}], "customName": "", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "useCustomName": false, "customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false}], "functions": {"correctPoints": {"type": "html", "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\n\n\nquestion.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\n//define ans as this \ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;", "parameters": [], "language": "javascript"}, "plotPoints": {"type": "html", "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n var ans;\n var nscope = new Numbas.jme.Scope([scope,{variables:{x:new Numbas.jme.types.TNum(0)}}]);\n//this is the beating heart of whatever plots the function,\n//I've changed this from being curve to functiongraph\n var line = board.create('functiongraph',[function(x){\nif(ans) {\n try {\nnscope.variables.x.value = x;\n var val = Numbas.jme.evaluate(ans,nscope).value;\n return val;\n }\n catch(e) {\nreturn 13;\n }\n}\nelse\n return 13;\n },-12,12]\n , {strokeColor:\"blue\",strokeWidth: 4} );\n \nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: false, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\nquestion.lines = {\n l:line, c:correct_line\n}\n\n question.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\nvar expr = question.parts[2].gaps[0].display.studentAnswer();\n\n//define ans as this \ntry {\n ans = Numbas.jme.compile(expr,scope);\n}\ncatch(e) {\n ans = null;\n}\nline.updateCurve();\ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;", "parameters": [], "language": "javascript"}}, "rulesets": {}, "type": "question"}, {"name": "Percentages and ratios - box of chocolates", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}], "type": "question", "statement": "A family receive a box of chocolates as a gift. There are five different kinds of chocolate inside: plain, nut, caramel, dark and coconut.
\nThe box contains equal numbers of each kind of chocolate..
", "advice": "100% represents the whole box of chocolates. As there are 5 different kinds of chocolate in the box and they are all represented equally, to calculate the percentage chocolates which are caramel, divide 100 by 5.
\nCaramel chocolate = $\\displaystyle\\frac{100}{5}$ = $20$% of the box.
\n\n\nb)
\nThe original number of chocolates in the box is stated. We worked out above that each type of chocolate makes up 20% of the box, so we need to work out 20% of {chocs}.
\nTo do this, either divide {chocs} by 100 and mulitply by 20, OR multiply {chocs} by 0.2. The two methods will give the same result.
\nMethod 1: $\\displaystyle\\frac{\\var{chocs}}{100}$ x $20$ = $\\var{type}$;
\nOR
\nMethod 2: $\\var{chocs}$ x $0.2$ = $\\var{type}$.
\n\n\nc)
\nThere are now {type} fewer chocolates in the box, but the remaining chocolates now represent 100% of the box. There are now only 4 types of chocolate in it and there is still equal representation inside the box.
\nUse the method from part a) to find out the equal share of each chocolate type.
\nEach type = $\\displaystyle\\frac{100}{4}$ = $25$% of the box.
\n\n\nd)
\ni)
\nThe first section asks you to compare plain chocolate and dark chocolate. It states that there are {p} plain chocolates and {d} dark chocolates left in the box.
\nInsert the numbers of each into the gaps.
\nPlain $\\var{p}$ : $\\var{d}$ Dark
\nFrom this, we should look to see if this answer can be simplified down. To do this, we need to find the greatest common divisor of $\\var{p}$ and $\\var{d}$.
\nThe greatest common divisor is $\\var{gcd}$.
\nUsing this value to simplify down the ratio by dividing each term by the value, the final answer is
\nPlain $\\var{ratio_plain}$ : $\\var{ratio_dark}$ Dark.
\nThis states that for every {ratio_plain} plain {if(ratio_plain=1,\"chocolate\",\"chocolates\")}, there {if(ratio_dark=1,\"is\",\"are\")} {ratio_dark} dark {if(ratio_dark=1,\"chocolate\",\"chocolates\")}.
\nTherefore, it is not possible to simplify further and the final answer is
\nPlain $\\var{p}$ : $\\var{d}$ Dark.
\nThis states that for every {p} plain {if(p=1,\"chocolate\",\"chocolates\")}, there {if(d=1,\"is\",\"are\")}{d} dark {if(d=1,\"chocolate\",\"chocolates\")}.
\n\nii)
\nThe second section asks you to compare coconut chocolates and the rest of the box. It states that there are {c} coconut chocolates. To calculate the number of chocolates in the rest of the box, add together the stated amounts of plain, dark and nutty chocolates:
\n$\\var{p}+\\var{d}+\\var{n}$ = $\\var{rob}$.
\nInsert these two figures into the gaps.
\nCoconut $\\var{c}$ : $\\var{rob}$ Other chocolates
\nFrom this, we should look to see if this answer can be simplified down. To do this, we need to find the greatest common divisor of $\\var{c}$ and $\\var{rob}$.
\nThe greatest common divisor is $\\var{gcd2}$.
\nUsing this value to simplify down the ratio by dividing each term by the value, the final answer is
\nCoconut $\\var{ratio_coconut}$ : $\\var{ratio_rest}$ Other chocolates.
\nThis states that for every {ratio_coconut} coconut {if(ratio_coconut=1,\"chocolate\",\"chocolates\")}, there {if(ratio_rest=1,\"is\",\"are\")} {ratio_rest} other {if(ratio_rest=1,\"chocolate\",\"chocolates\")} in the box.
\nTherefore, it is not possible to simplify further and the final answer is
\nCoconut $\\var{c}$ : $\\var{rob}$ Other chocolates.
\nThis states that for every {c} coconut {if(c=1,\"chocolate\",\"chocolates\")}, there {if(rob=1,\"is\",\"are\")} {rob} other {if(rob=1,\"chocolate\",\"chocolates\")} in the box.
", "variables": {"ratio_dark": {"templateType": "anything", "name": "ratio_dark", "definition": "d/gcd(p,d)", "description": "Number of dark chocolates in ratio of plain to dark.
", "group": "Ungrouped variables"}, "ratio_rest": {"templateType": "anything", "name": "ratio_rest", "definition": "rob/gcd(c, rob)", "description": "Number of 'rest of box' chocolates in ratio of coconut to rest of box.
", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "name": "d", "definition": "random(1..3)*p", "description": "Number of dark chocolates on day 3.
", "group": "Ungrouped variables"}, "gcd2": {"templateType": "anything", "name": "gcd2", "definition": "gcd(c,rob)", "description": "", "group": "Ungrouped variables"}, "type": {"templateType": "anything", "name": "type", "definition": "chocs/5", "description": "Number of each type of chocolate in the box.
", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "definition": "random(1..14 except 7 except 11 except 13)", "description": "Number of coconut chocolates on day 3.
", "group": "Ungrouped variables"}, "chocs": {"templateType": "randrange", "name": "chocs", "definition": "random(70..95#5)", "description": "Total number of chocolates in the box before eating.
", "group": "Ungrouped variables"}, "rob": {"templateType": "anything", "name": "rob", "definition": "p+n+d", "description": "Sum of the rest of the box excluding coconut.
", "group": "Ungrouped variables"}, "p": {"templateType": "anything", "name": "p", "definition": "random(2..5)", "description": "Number of plain chocolates on day 3.
", "group": "Ungrouped variables"}, "ratio_plain": {"templateType": "anything", "name": "ratio_plain", "definition": "p/gcd(p,d)", "description": "Number of plain chocolates in ratio of plain to dark.
", "group": "Ungrouped variables"}, "prob": {"templateType": "anything", "name": "prob", "definition": "precround({n/{a},2)", "description": "Probability that a nutty chocolate is selected from the box on day 3.
", "group": "Ungrouped variables"}, "gcd": {"templateType": "anything", "name": "gcd", "definition": "gcd(p,d)", "description": "", "group": "Ungrouped variables"}, "n": {"templateType": "anything", "name": "n", "definition": "random(1..14 except 7 except 11 except 13)", "description": "Number of nutty chocolates on day 3.
", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "name": "a", "definition": "p+n+d+c", "description": "Number of chocolates in the box on day 3.
", "group": "Ungrouped variables"}, "perc": {"templateType": "anything", "name": "perc", "definition": "100*(prob)", "description": "Percentage version of probability.
", "group": "Ungrouped variables"}, "minusc": {"templateType": "anything", "name": "minusc", "definition": "{chocs-type}", "description": "Number of chocolates in the box minus caramel.
", "group": "Ungrouped variables"}, "ratio_coconut": {"templateType": "anything", "name": "ratio_coconut", "definition": "c/gcd(c, rob)", "description": "Number of coconut chocolates in ratio of coconut to rest of box.
", "group": "Ungrouped variables"}}, "functions": {}, "tags": ["percentages", "ratios", "Ratios", "taxonomy"], "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "20", "showFeedbackIcon": true, "minValue": "20", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "What percentage of the box of chocolates is represented by the caramel chocolates?
\nCaramel chocolate = [[0]] % of the box.
", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "type", "showFeedbackIcon": true, "minValue": "type", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "If there were $\\var{chocs}$ chocolates in the box originally, how many of each kind were there?
\nThere are [[0]] of each type of chocolate in the box.
\n", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "25", "showFeedbackIcon": true, "minValue": "25", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "Caramel flavoured chocolate is the family favourite, and so all of these chocolates are eaten first, and none of the other kinds are touched.
\nWhat percentage of the remaining chocolates are plain?
\nPlain chocolates = [[0]]% of the box.
", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ratio_plain", "showFeedbackIcon": true, "minValue": "ratio_plain", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ratio_dark", "showFeedbackIcon": true, "minValue": "ratio_dark", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ratio_coconut", "showFeedbackIcon": true, "minValue": "ratio_coconut", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ratio_rest", "showFeedbackIcon": true, "minValue": "ratio_rest", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "Over the next few days, the remaining chocolates in the box are slowly devoured so that by day three, all that remain are:
\n$\\var{p}$ plain chocolates, $\\var{n}$ nutty chocolates, $\\var{c}$ coconut chocolates and $\\var{d}$ dark chocolates.
\n\ni) What is the ratio of plain to dark chocolates? Give your answer in its simplest form.
\nPlain [[0]] : [[1]] Dark
\n\nii) What is the ratio of coconut chocolates to the rest of the box? Give your answer in its simplest form.
\nCoconut [[2]] : [[3]] Rest of the box
\n", "type": "gapfill"}], "ungrouped_variables": ["chocs", "type", "p", "n", "d", "c", "rob", "prob", "a", "perc", "minusc", "ratio_plain", "ratio_dark", "ratio_coconut", "ratio_rest", "gcd", "gcd2"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "A simple situational question about a box of chocolates, asking how many of each type there are, what percentage of the box they represent, the probability of picking one and ratios of different types.
"}, "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Substitute values into formulas", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "metadata": {"description": "Substitute given values into formulas.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["r", "x1", "n", "x2", "const", "sales"], "type": "question", "advice": "When inserting numbers into your calculator make sure you place brackets correctly.
\nAs $x = \\var{n+2}$,
\nsubstitute $\\var{n+2}$ into $\\var{x2}x^2 + \\var{x1}x + \\var{const}$.
\n\\begin{align}
\\var{x2}x^2 + \\var{x1}x + \\var{const} &= \\var{x2} (\\var{n+2})^2 + \\var{x1}(\\var{n+2}) + \\var{const} \\\\
&= \\simplify{{x2} ({n+2})^2 + {x1}({n+2}) + {const}}\\,.
\\end{align}
b)
\nAs $y = \\var{n}$,
\nsubstitute $\\var{n}$ into $\\var{n+1}y^2-\\var{x2}y$.
\n\\begin{align}
\\var{n+1}y^2-\\var{x2}y &= \\var{n+1}(\\var{n})^2-\\var{x2}(\\var{n}) \\\\
&= \\simplify{{n+1}({n})^2-{x2}({n})}\\,.
\\end{align}
c)
As we are given a temperature in degrees Celcius, $T_C = \\var{T_C}°C.$
\nSubstituting $T_C$ into $T_C = 1.8\\,T_C + 32$.
\n\\begin{align}
T_F &=1.8\\, T_C+32 \\\\
&=1.8 (\\var{T_C}) + 32 \\\\
&= \\var{dpformat(1.8 {T_C} +32, 1)}\\,°F\\,.
\\end{align}
Substitute the given values in the equations below.
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "A curve is defined by a function $y=\\simplify{{x2}x^2 + {x1}x + {const}}$.
\nWhat is the $y$ coordinate value of the point on the curve at $x=\\var{n+2}$?
\n$y =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{n+1}{n}^2-{x2}{n}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{n+1}{n}^2-{x2}{n}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "{name[n]} sells luxury yachts.
\nThe predicted sales of the luxury yachts are defined by
\n\\[S=\\simplify{{n+1}y^2-{x2}y},\\]
\nwhere
$S$ is the number of sales predicted this year;
$y$ is the number of luxury yachts sold in the previous year.
{pronoun} sold {n} yachts in the previous year.
\nCalculate $S$, the number of sales predicted this year.
\n$S =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "T_F", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "T_F", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "You can convert temperatures from degrees celsius to degrees fahrenheit by using the formula
\n\\[T_F=1.8\\, T_C+32,\\]
\nwhere
$T_F$ = Temperature in $°F$
$T_C$ = Temperature in $°C$.
Convert $\\var{T_C}°C$ into degrees fahrenheit.
\n$T_F =$ [[0]] $°F$
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", "definition": "random(0..4#1)", "group": "Ungrouped variables", "name": "n", "templateType": "anything"}, "sales": {"description": "", "definition": "(n+1)n^2-x2*n", "group": "Ungrouped variables", "name": "sales", "templateType": "anything"}, "const": {"description": "The constant coefficient
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", "definition": "T_C*1.8+32", "group": "Temperature conversion", "name": "T_F", "templateType": "anything"}, "r": {"description": "A random variable which will be inputted by the student.
", "definition": "random(1..50#0.1)", "group": "Ungrouped variables", "name": "r", "templateType": "anything"}, "x2": {"description": "The x^2 coefficient
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", "definition": "random(1..50)", "group": "Ungrouped variables", "name": "x1", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "showstudentname": true, "name": "Year 10 Practice Exam 1", "type": "exam", "contributors": [{"name": "Neil Anderson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2916/"}], "extensions": ["geogebra", "jsxgraph", "stats"], "custom_part_types": [], "resources": []}