// Numbas version: exam_results_page_options {"showQuestionGroupNames": false, "duration": 0, "name": "Units of measurement [L1 Randomised]", "metadata": {"description": "

A collection of questions on working with units of measurement, mainly in the SI/metric system.

Several 'real-world' examples.

", "licence": "Creative Commons Attribution 4.0 International"}, "feedback": {"allowrevealanswer": true, "showanswerstate": true, "advicethreshold": 0, "feedbackmessages": [], "showactualmark": true, "intro": "", "showtotalmark": true}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}, "allowPause": true}, "navigation": {"reverse": true, "showresultspage": "oncompletion", "browse": true, "preventleave": true, "allowregen": true, "showfrontpage": true, "onleave": {"action": "none", "message": ""}}, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 1, "name": "Group", "questions": [{"name": "Use formulae for the area and volume of geometric shapes", "extensions": ["geogebra"], "custom_part_types": [], "resources": [["question-resources/icecrea_QqVaCIf.svg", "/srv/numbas/media/question-resources/icecrea_QqVaCIf.svg"], ["question-resources/frisbee_variable_TESZa4J.svg", "/srv/numbas/media/question-resources/frisbee_variable_TESZa4J.svg"], ["question-resources/tennis-ball_with_variable_MBOLQeM.svg", "/srv/numbas/media/question-resources/tennis-ball_with_variable_MBOLQeM.svg"]], "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/"}], "tags": ["Area", "area", "area of a circle", "area of a trapezium", "area of a triangle", "Area of a triangle", "Circle", "circle", "cone", "Cone", "geometry", "taxonomy", "trapezium", "triangle", "Triangle", "volume", "Volume", "volume of a cone", "volume of a sphere"], "metadata": {"description": "

Substitute values into formulae for the area or volume of various geometric objects.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Answer the following questions by substituting the correct values into the given equations.

", "advice": "

When inserting numbers into your calculator, make sure that you place brackets correctly.

a)

We can see from the diagram that the radius of the frisbee is $\\var{mccall[2]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the area of a circle with $\\var{mccall[2]}$ gives,

\\begin{align}
\\mathrm{Area} &= \\pi r^2 \\\\
&= \\pi\\times(\\var{mccall[2]})^2 \\\\
&= \\var{dpformat((mccall[2])^2, 2)}\\pi\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(pi *(mccall[2])^2, 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}

b)

We can see from the diagram that the triangle has two sides with lengths $\\var{length_cdp2}$ $\\mathrm{cm}$, $\\var{length_bdp2}$ $\\mathrm{cm}$ and an angle $\\var{c_thetadp2}\\mathrm{°}$ .
Replacing the letters $a$, $b$ and $C$ in the formula for the area of a triangle with $\\var{length_cdp2}$, $\\var{length_bdp2}$ and $\\var{c_thetadp2}$ respectively gives,

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2}ab\\sin{C} \\\\
&= \\frac{1}{2} \\times \\var{length_cdp2} \\times \\var{length_bdp2} \\times \\sin(\\var{c_thetadp2}) \\\\
&= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2* pi/180), 5)}\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(0.5*(length_cdp2)*(length_bdp2)*sin(c_thetadp2 * pi/180), 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}

c)

We can see from the diagram that the radius of the cone is $\\var{r}$ $\\mathrm{cm}$ and the height is $\\var{h}$ $\\mathrm{cm}$.
Replacing the letters $r$ and $h$ in the formula for the volume of a cone with $\\var{r}$ $\\mathrm{cm}$ and $\\var{h}$ $\\mathrm{cm}$ respectively gives,

\\begin{align}
\\mathrm{Volume} &= \\frac{h}{3} \\pi r^2 \\\\
&= \\frac{(\\var{h})}{3} \\times \\pi \\times (\\var{r})^2 \\\\
&= \\var{dpformat((pi)*(h/3)*(r)^2 , 5)}\\, \\mathrm{cm}^3 \\\\
&=\\var{dpformat(h/3 * pi * (r)^2, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

d)

We can see from the diagram that the radius of the tennis ball is $\\var{mccall[1]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the volume of a sphere with $\\var{mccall[1]}$ gives,

\\begin{align}
\\mathrm{Volume} &= \\frac{4}{3} \\pi r^3 \\\\
&= \\frac{(4)}{(3)} \\times \\pi \\times (\\var{mccall[1]})^3 \\\\
&= \\var{dpformat((4/3)*pi*mccall[1]^3, 5)}\\, \\mathrm{cm}^3 \\\\
&= \\var{precround(((4/3)* pi) *(mccall[1])^3, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

e)

We can see from the diagram that the trapezium has two parallel sides with length $\\var{trap_length_a}$ $\\mathrm{cm}$, $\\var{trap_length_b}$ $\\mathrm{cm}$ and height $\\var{trap_h}$ $\\mathrm{cm}$.
Replacing the letters $a$, $b$ and $h$ in the formula for the area of a trapezium with $\\var{trap_length_a}$, $\\var{trap_length_b}$ and $\\var{trap_h}$ respectively gives,

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5)(trap_length_a +trap_length_b) trap_h, 2)}\\, \\mathrm{cm}^2 \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"length_cdp2": {"name": "length_cdp2", "group": "Triangle variables", "definition": "precround(sqrt((b[0]-c[0])^2+(b[1]-c[1])^2),2)", "description": "

Rounded value for the length of c.

", "templateType": "anything", "can_override": false}, "trap_e": {"name": "trap_e", "group": "Trapezium variables", "definition": "vector(trap_b[0],-4)", "description": "

Defines the point for the height of the trapezium.

", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Cone variables", "definition": "random(3..6#0.1)", "description": "

A random variable which will be inputted by the student.

", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "Quadratic variables", "definition": "random(1..100#1)", "description": "

The constant coefficient

", "templateType": "anything", "can_override": false}, "name": {"name": "name", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "description": "

List of names to randomise. Can change to any name inserted

", "templateType": "anything", "can_override": false}, "length_bdp2": {"name": "length_bdp2", "group": "Triangle variables", "definition": "precround(sqrt((a[0]-c[0])^2+(a[1]-c[1])^2),2)", "description": "

Rounded value for the length of b.

", "templateType": "anything", "can_override": false}, "length_c": {"name": "length_c", "group": "Triangle variables", "definition": "sqrt((b[0]-c[0])^2+(b[1]-c[1])^2)", "description": "

For triangle - The length of the vector BC

", "templateType": "anything", "can_override": false}, "trap_length_b": {"name": "trap_length_b", "group": "Trapezium variables", "definition": "sqrt((trap_d[0]-trap_a[0])^2+(trap_d[1]-trap_a[1])^2)", "description": "", "templateType": "anything", "can_override": false}, "c_thetadp2": {"name": "c_thetadp2", "group": "Triangle variables", "definition": "precround((180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))),2)", "description": "

Rounded theta value.

", "templateType": "anything", "can_override": false}, "trap_length_a": {"name": "trap_length_a", "group": "Trapezium variables", "definition": "sqrt((trap_c[0]-trap_b[0])^2+(trap_c[1]-trap_b[1])^2)", "description": "", "templateType": "anything", "can_override": false}, "length_b": {"name": "length_b", "group": "Triangle variables", "definition": "sqrt((a[0]-c[0])^2+(a[1]-c[1])^2)", "description": "

For triangle - The length of the vector AC

", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Triangle variables", "definition": "vector(2,0)", "description": "

Position of point B in Geogebra. This point is randomised to make the triangles different.

", "templateType": "anything", "can_override": false}, "pronoun": {"name": "pronoun", "group": "Name variables", "definition": "if(mod(n,2)=0,\"he\",\"she\")", "description": "

Defines the pronoun in the question.

", "templateType": "anything", "can_override": false}, "trap_defs": {"name": "trap_defs", "group": "Trapezium variables", "definition": "[\n ['A', trap_a],\n ['B', trap_b],\n ['C', trap_c],\n ['D', trap_d],\n ['E', trap_e]\n ]", "description": "

Definition of the points to put into Geogebra

", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Cone variables", "definition": "random(11..17#0.1)", "description": "

The height for volume of a cone.

", "templateType": "anything", "can_override": false}, "trap_h": {"name": "trap_h", "group": "Trapezium variables", "definition": "trap_b[1] + 4", "description": "

Height of the trapezium

", "templateType": "anything", "can_override": false}, "Triangle_area": {"name": "Triangle_area", "group": "Triangle variables", "definition": "1/2*(length_cdp2)(length_bdp2)(sin(c_thetadp2 * pi/180))", "description": "

This calculates the area of the triangle for part b)

", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Triangle variables", "definition": "vector(-3,0)", "description": "

Position of the point A in Geogebra. This point is fixed so the triangle doesn't hang in one corner or the whole page.

", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Name variables", "definition": "random(0..4#1)", "description": "

n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.

", "templateType": "anything", "can_override": false}, "name2": {"name": "name2", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "description": "

List of names to randomise. Can change to any name inserted

", "templateType": "anything", "can_override": false}, "trap_rand": {"name": "trap_rand", "group": "Trapezium variables", "definition": "random(1..3#1)", "description": "

A random number to define the height of the trapezium.

", "templateType": "anything", "can_override": false}, "c_theta": {"name": "c_theta", "group": "Triangle variables", "definition": "(180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))) ", "description": "

Theta is randomised by the lengths

", "templateType": "anything", "can_override": false}, "trap_areadp1": {"name": "trap_areadp1", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h,1)", "description": "

Calculates the area of the trapezium

", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Triangle variables", "definition": "[\n ['A',a],\n ['B',b],\n ['C',c]\n ]", "description": "

Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.

", "templateType": "anything", "can_override": false}, "length_a": {"name": "length_a", "group": "Triangle variables", "definition": "sqrt((a[0]-b[0])^2+(a[1]-b[1])^2)", "description": "

For triangle - The length of the vector AB

", "templateType": "anything", "can_override": false}, "trap_d": {"name": "trap_d", "group": "Trapezium variables", "definition": "vector(random(5..7#0.1), -4)", "description": "

Creates the point D on the trapezium

", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Triangle variables", "definition": "vector(\n random(2..5#0.1),\n random(2..5#0.1)\n )", "description": "

Triangle - A variable point which ultimately decides how the triangle looks.

", "templateType": "anything", "can_override": false}, "trap_a": {"name": "trap_a", "group": "Trapezium variables", "definition": "vector(1,-4)", "description": "

Creates the point A on the trapezium

", "templateType": "anything", "can_override": false}, "trap_c": {"name": "trap_c", "group": "Trapezium variables", "definition": "vector(random(4..5.5#0.1), trap_rand)", "description": "

Creates the point C on the trapezium

", "templateType": "anything", "can_override": false}, "mccall": {"name": "mccall", "group": "RNG", "definition": "[0,random(3.1..3.7#0.1),random(5..20#0.1)]\n", "description": "

Matrix of random variables used to create length in the questions.

", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Quadratic variables", "definition": "random(1..10#1)", "description": "

The x^2 coefficient

", "templateType": "anything", "can_override": false}, "trap_areadp2": {"name": "trap_areadp2", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h, 2)", "description": "", "templateType": "anything", "can_override": false}, "trap_b": {"name": "trap_b", "group": "Trapezium variables", "definition": "vector(random(1.5..2.5#0.1), trap_rand)", "description": "

Creates the point B on the trapezium

", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Quadratic variables", "definition": "random(1..50)", "description": "

The x coefficient

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Trapezium variables", "variables": ["trap_a", "trap_b", "trap_c", "trap_d", "trap_h", "trap_rand", "trap_defs", "trap_length_a", "trap_length_b", "trap_e", "trap_areadp1", "trap_areadp2"]}, {"name": "Triangle variables", "variables": ["Triangle_area", "c", "b", "a", "c_theta", "length_a", "length_b", "length_c", "defs", "length_bdp2", "length_cdp2", "c_thetadp2"]}, {"name": "Name variables", "variables": ["name", "name2", "pronoun", "n"]}, {"name": "Quadratic variables", "variables": ["x2", "x1", "const"]}, {"name": "Cone variables", "variables": ["r", "h"]}, {"name": "RNG", "variables": ["mccall"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the area of a frisbee, assuming that the frisbee can be modelled as a circle, given the formula for the area of a circle is

\\[\\mathrm{Area} = \\pi r^2.\\]

$\\mathrm{Area}$ = [[0]] $\\mathrm{cm}^2$ Round your answer to 2 decimal places.

", "gaps": [{"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, "minValue": "pi*{mccall[2]}^2 - 0.05", "maxValue": "pi*{mccall[2]}^2 + 0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the area of the triangle given that the area of any triangle can be calculated using the formula

\\[\\mathrm{Area} = \\frac{1}{2}ab\\sin{C}.\\]

{geogebra_applet('https://www.geogebra.org/m/jcUJu6F4',defs)}

All lengths are in centimetres.

$\\mathrm{Area} =$ [[0]] $\\mathrm{cm}^2$ Round your answer to 2 decimal places.

", "gaps": [{"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, "minValue": "Triangle_area-0.05", "maxValue": "Triangle_area+0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the volume of a cone given the formula for the volume of a cone is

\\[\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\\]

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.

", "gaps": [{"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, "minValue": "{(h/3)}*{r}^2*pi -0.5", "maxValue": "{(h/3)}*{r}^2*pi +0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

{name[n]} has a tennis ball and {pronoun} wants to find the volume of the ball. Using the diagram and the formula for the volume of a sphere, calculate the volume of the ball.

\\[\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\\]

$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.

", "gaps": [{"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, "minValue": "4/3 * pi * {mccall[1]}^3 - 0.5", "maxValue": "4/3 * pi * {mccall[1]}^3 + 0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the area of the trapezium given the formula for the area of a trapezium is

\\[\\mathrm{Area} = \\frac{1}{2}(a+b) h .\\]

{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}

All lengths are given in metres.

$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$ Round your answer to 1 decimal place.

", "gaps": [{"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, "minValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} - 0.1", "maxValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} + 0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Find bounds for distance and time spent running, given imprecise measurements", "extensions": ["random_person"], "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": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "metadata": {"description": "

Multiplication and division of upper and lower bounds.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["speed", "distance", "time", "atime", "person", "pronouns", "verbs"], "statement": "

{person['name']} is a keen runner. {capitalise(pronouns['they'])} run{verbs} at an average speed of {speed}km/h, rounded to the nearest integer.

", "variable_groups": [], "rulesets": {}, "type": "question", "advice": "

We're not certain about some of the measurements given in this question - we only know the rounded values. This means that the true value could be lower or higher than the given measurement.

We can find upper and lower bounds for the given measurements. Any values we go on to calculate will also be uncertain and have upper and lower bounds.

To find bounds for a given measurement, we divide the degree of accuracy by 2 and subtract or add this to our estimate to get lower and upper bounds respectively.

For example, $52$ rounded to the nearest integer has a lower bound of $51.5$ and an upper bound of $52.5$.

a)

The distance travelled is given by

\\[ d = \\text{Average speed} \\times \\text{Time taken} \\]

We find bounds for speed and time first.

Lower bound for speed: $\\var{speed} - 0.5 = \\var{speed - 0.5} \\text{ km/h}$

Upper bound for speed: $\\var{speed} + 0.5 = \\var{speed + 0.5} \\text{ km/h}$

First, note that the speed is given in km/h and we want to find the distance in km. We will convert the given time into hours.

Lower bound for time taken:

\\begin{align} \\var{atime} - 0.5 &= \\var{atime - 0.5} \\text{ min} \\\\[0.5em]
&= \\frac{\\var{atime - 0.5}}{60} \\text{ h}
\\end{align}

Upper bound for time taken:

\\begin{align} \\var{atime} + 0.5 &= \\var{atime + 0.5} \\text{ min} \\\\[0.5em]
&= \\frac{\\var{atime + 0.5}}{60} \\text{ h}
\\end{align}

Since we're multiplying the speed and time together, the lower bound for distance is the slowest speed multiplied by the shortest time:

\\begin{align}
\\text{Lower bound} &= \\text{lower bound for speed} \\times \\text{lower bound for time}\\\\
&= \\var{speed - 0.5} \\times \\frac{\\var{(atime - 0.5)}}{60} \\\\
&= \\var{precround((speed-0.5)*(atime - 0.5)/60, 2 )} \\text{ km} \\quad \\text{(rounded to 2 decimal places).}
\\end{align}

The upper bound for distance is the fastest speed multiplied by the longest time:

\\begin{align}
\\text{Upper bound} &= \\text{upper bound for speed} \\times \\text{upper bound for time} \\\\
&= \\var{speed + 0.5} \\times \\frac{\\var{atime + 0.5}}{60} \\\\
&= \\var{precround((speed+0.5)*(atime + 0.5)/60, 2 )} \\text{ km} \\quad \\text{(rounded to 2 decimal places).}
\\end{align}

Hence,

\\[\\var{precround((speed-0.5)*(atime - 0.5)/60, 2 )} \\leq d \\lt \\var{precround((speed+0.5)*(atime + 0.5)/60, 2 )} \\text{.}\\]

b)

We're told the speed and the distance travelled, so the time taken is given by

\\[ t = \\frac{\\text{Distance travelled}}{\\text{Average speed}} \\]

We found upper and lower bounds for {person['name']}'s average speed above.

The distance of the evening run is given to the nearest kilometre, so we can compute bounds as follows:

Lower bound for distance: $\\var{distance} - 0.5 = \\var{distance - 0.5} \\mathrm{km}$

Upper bound for distance: $\\var{distance} + 0.5 = \\var{distance + 0.5} \\mathrm{km}$

The upper bound for the time taken is the longest distance divided by the slowest speed:

\\begin{align}
\\text{Upper bound} &= \\text{upper bound for distance} \\div \\text{lower bound for speed} \\\\
&= \\var{distance + 0.5} \\div \\var{speed - 0.5} \\\\
&= \\var{(distance + 0.5)/(speed - 0.5)} \\text{ hours.}
\\end{align}

We're asked for the answer in minutes, to two decimal places.

\\begin{align}
\\var{(distance + 0.5)/(speed - 0.5)} \\text{ hours} &= \\var{(distance + 0.5)/(speed - 0.5)}\\times 60 \\text{ min} \\\\
&= \\var{precround((distance + 0.5)/(speed - 0.5)*60, 2)} \\text{ minutes} \\quad \\text{(rounded to 2 decimal places)}
\\end{align}

The lower bound for time is the shortest distance divided by the fastest speed:

\\begin{align}
\\text{Lower bound} &= \\text{lower bound for distance} \\div \\text{upper bound for speed} \\\\
&= \\var{distance - 0.5} \\div \\var{speed + 0.5} \\\\
&= \\var{(distance - 0.5)/(speed + 0.5)} \\text{ hours.}
\\end{align}

Converting into minutes, to two decimal places:

\\begin{align}
\\var{(distance - 0.5)/(speed + 0.5)} \\text{ hours} &= \\var{(distance - 0.5)/(speed + 0.5)}\\times 60 \\text{ min} \\\\
&= \\var{precround((distance - 0.5)/(speed + 0.5)*60, 2)} \\text{ min.}
\\end{align}

Therefore, we cannot confidently say {person['name']}'s time was less than {time*60 +1} minutes as the upper bound for {pronouns['their']} time, $\\var{precround((distance + 0.5)/(speed - 0.5)*60, 2)}$ minutes, is above this threshold.

", "parts": [{"scripts": {}, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "prompt": "

We're not certain about some of the measurements given in this question - we only know the rounded values. This means that the true value could be lower or higher than the given measurement.

Compute upper and lower bounds for {person['name']}'s average speed and the time spent running, then use those to find upper and lower bounds for the distance travelled.

", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0}], "type": "gapfill", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "prompt": "

Suppose {person['name']} ran for {atime} minutes, rounded to the nearest minute.

Using the rounded figures for {pronouns['their']} average speed and time spent running, calculate upper and lower bounds for the distance, $d$, that {person['name']} ran.

Round your answers to two decimal places.

[[0]] $\\leq d \\lt$ [[1]]km

", "gaps": [{"correctAnswerFraction": false, "variableReplacements": [], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "2", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "{speed - 0.5}*{(atime - 0.5)/60}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{speed - 0.5}*{(atime - 0.5)/60}", "mustBeReduced": false, "allowFractions": false, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "variableReplacements": [], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "2", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "{speed + 0.5}*{(atime + 0.5)/60}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{speed + 0.5}*{(atime + 0.5)/60}", "mustBeReduced": false, "allowFractions": false, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "showCorrectAnswer": true}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "

Now consider {person['name']}'s evening run. {capitalise(pronouns['they'])} covered a distance of {precround(distance,0)}km, rounded to the nearest kilometre.

{capitalise(pronouns['their'])} friend's record time for the evening run is exactly {time*60 +1} minutes.

Can we confidently say that {person['name']} beat {pronouns['their']} friend's record?

First, calculate the lower and upper bounds for {person['name']}'s time, $t$.

Round your answers to two decimal places.

[[0]] $\\leq t \\lt $ [[1]] minutes

", "gaps": [{"correctAnswerFraction": false, "variableReplacements": [], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "2", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "({precround(distance,0) - 0.5}/{speed + 0.5})*60", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "({precround(distance,0) - 0.5}/{speed + 0.5})*60", "mustBeReduced": false, "allowFractions": false, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "variableReplacements": [], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "2", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "({precround(distance,0) + 0.5}/{speed - 0.5})*60", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "({precround(distance,0) + 0.5}/{speed - 0.5})*60", "mustBeReduced": false, "allowFractions": false, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "showCorrectAnswer": true}], "showFeedbackIcon": true, "marks": 0}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "prompt": "

Therefore, we [[0]] confidently say {pronouns['their']} time was less than {time*60+1} minutes as the [[1]] bound for time is [[2]] the given threshold of {time*60+1} mins.

", "gaps": [{"shuffleChoices": false, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

can

", "

cannot

"], "distractors": ["", ""], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}, {"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

upper

", "

lower

"], "distractors": ["", ""], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}, {"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["

above

", "

below

", "

equal to

"], "distractors": ["", "", ""], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0, 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "marks": 0}], "tags": ["division", "limits of accuracy", "lower bounds", "multiplication", "random names", "taxonomy", "upper bounds"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"pronouns": {"description": "", "definition": "person['pronouns']", "group": "Ungrouped variables", "name": "pronouns", "templateType": "anything"}, "distance": {"description": "

Distance in kms.

", "definition": "speed*time", "group": "Ungrouped variables", "name": "distance", "templateType": "anything"}, "atime": {"description": "

Time in minutes

", "definition": "random(5..30 #5)", "group": "Ungrouped variables", "name": "atime", "templateType": "anything"}, "verbs": {"description": "", "definition": "if(person['gender']='neutral','','s')", "group": "Ungrouped variables", "name": "verbs", "templateType": "anything"}, "person": {"description": "", "definition": "random_person()", "group": "Ungrouped variables", "name": "person", "templateType": "anything"}, "time": {"description": "

Time in hours.

", "definition": "random(0.25..1 #0.25)", "group": "Ungrouped variables", "name": "time", "templateType": "anything"}, "speed": {"description": "

Speed in km/h.

", "definition": "random(7..11)", "group": "Ungrouped variables", "name": "speed", "templateType": "anything"}}, "variablesTest": {"maxRuns": "1000", "condition": ""}}, {"name": "Rounding and estimating calculations - painting a room", "extensions": ["random_person"], "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/"}], "type": "question", "statement": "

{person['name']} just bought a new house. {capitalise(pronouns['their'])} new bedroom's wall and ceiling are currently painted white, but {pronouns['they']} would like to paint these {colour}.

The dimensions of the floor are $\\var{length}\\,\\mathrm{m} \\times \\var{width}\\,\\mathrm{m}$ and the room is $\\var{height}\\,\\mathrm{m}$ high.

{person['name']} want{verbs} to know how much paint to buy so {pronouns['they']} can paint all four walls and the ceiling {colour}.

", "variablesTest": {"condition": "mod(rall,15)<>0", "maxRuns": 100}, "variables": {"buckets": {"description": "", "name": "buckets", "group": "Calculations", "templateType": "anything", "definition": "ceil(rall/bucket_area)"}, "bucket_area": {"description": "

The area that can be painted with one bucket of paint.

", "name": "bucket_area", "group": "Calculations", "templateType": "anything", "definition": "15"}, "colour": {"description": "", "name": "colour", "group": "Random bits", "templateType": "anything", "definition": "random(\"green\", \"red\", \"orange\", \"yellow\", \"blue\", \"purple\", \"pink\")"}, "width": {"description": "", "name": "width", "group": "Random bits", "templateType": "anything", "definition": "random(2.50..5.00 #0.1) + random(0.01..0.09 #0.01)"}, "w": {"description": "", "name": "w", "group": "Calculations", "templateType": "anything", "definition": "ceil(width)"}, "verbs": {"description": "", "name": "verbs", "group": "Person", "templateType": "anything", "definition": "if(person['gender']='neutral','','s')"}, "height": {"description": "", "name": "height", "group": "Random bits", "templateType": "anything", "definition": "random(2.10..2.70 #0.1) + random(0.01..0.09 #0.01)"}, "h": {"description": "", "name": "h", "group": "Calculations", "templateType": "anything", "definition": "ceil(height)"}, "rall": {"description": "", "name": "rall", "group": "Calculations", "templateType": "anything", "definition": "rceiling + rwall1*2 + rwall2*2"}, "pronouns": {"description": "", "name": "pronouns", "group": "Person", "templateType": "anything", "definition": "person['pronouns']"}, "length": {"description": "", "name": "length", "group": "Random bits", "templateType": "anything", "definition": "random(3.50..8.00 #0.1) + random(0.01..0.09 #0.01)"}, "person": {"description": "", "name": "person", "group": "Person", "templateType": "anything", "definition": "random_person()"}, "rwall1": {"description": "", "name": "rwall1", "group": "Calculations", "templateType": "anything", "definition": "l*h"}, "rwall2": {"description": "", "name": "rwall2", "group": "Calculations", "templateType": "anything", "definition": "w*h"}, "rceiling": {"description": "", "name": "rceiling", "group": "Calculations", "templateType": "anything", "definition": "w*l"}, "l": {"description": "", "name": "l", "group": "Calculations", "templateType": "anything", "definition": "ceil(length)"}}, "functions": {}, "tags": ["random names", "taxonomy"], "variable_groups": [{"name": "Random bits", "variables": ["colour", "height", "width", "length"]}, {"name": "Person", "variables": ["person", "pronouns", "verbs"]}, {"name": "Calculations", "variables": ["h", "w", "l", "rwall1", "rwall2", "rceiling", "rall", "bucket_area", "buckets"]}], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "choices": ["

Overestimate and therefore we round each measurement up.

", "

Underestimate and therefore we round each measurement down.

"], "showFeedbackIcon": true, "shuffleChoices": false, "displayColumns": "1", "variableReplacements": [], "marks": 0, "matrix": ["1", 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "

Is it better to overestimate or underestimate in this situation?

[[0]]

", "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "rall", "showFeedbackIcon": true, "minValue": "rall", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "stepsPenalty": 0, "prompt": "

Rounding each measurement to the nearest metre, estimate the whole area to be painted {colour}.

[[0]] m²

", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "

The room is {length}m long, {width}m wide, and {height}m high.

Round each measurement in the direction you decided on above.

", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(length)", "showFeedbackIcon": true, "prompt": "

Round the length to the nearest metre.

", "minValue": "ceil(length)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(width)", "showFeedbackIcon": true, "prompt": "

Round the width to the nearest metre.

", "minValue": "ceil(width)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "maxValue": "ceil(height)", "showFeedbackIcon": true, "prompt": "

Round the height to the nearest metre.

", "minValue": "ceil(height)", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "0.1", "showCorrectAnswer": true}], "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "buckets", "showFeedbackIcon": true, "minValue": "buckets", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "

One bucket of {colour} paint is enough to paint an area of 15m². How many buckets should {person['name']} buy to ensure {pronouns['they']} {if(person['gender']='neutral','have','has')} enough paint?

[[0]]

", "marks": 0}], "ungrouped_variables": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Estimate the number of buckets of paint to buy, by rounding measurements of a room up to the nearest metre and estimating the total area.

"}, "preamble": {"css": "", "js": ""}, "advice": "

a)

It is much better to have spare paint than not to have enough of it. So it is better to overestimate the area.

Therefore, we round each measurement up.

b)

We round each of our measurements up to the nearest whole metre:

Length: $\\var{length}\\,\\mathrm{m} \\approx \\var{l}\\,\\mathrm{m}$.

Width: $\\var{width}\\,\\mathrm{m} \\approx \\var{w}\\,\\mathrm{m}$.

Height: $\\var{height}\\,\\mathrm{m} \\approx \\var{h}\\,\\mathrm{m}$.

The total area consists of five areas: two walls of $\\var{l}\\,\\text{m} \\times \\var{h}\\,\\text{m}$ (length by height); two walls of $\\var{w}\\,\\text{m} \\times \\var{h}\\,\\text{m}$ (width by height); and a ceiling of $\\var{l}\\,\\text{m} \\times \\var{w}\\,\\text{m}$ (length by width).

\\[ \\begin{align}
\\var{l}\\,\\text{m} \\times \\var{h}\\,\\text{m} &= \\var{l*h}\\,\\text{m}^2
\\\\ \\var{w}\\,\\text{m} \\times \\var{h}\\,\\text{m} &= \\var{w*h}\\,\\text{m}^2
\\\\ \\var{l}\\,\\text{m} \\times \\var{w}\\,\\text{m} &= \\var{l*w}\\,\\text{m}^2
\\end{align}\\]

Therefore, the total area {person['name']} needs to paint is

\\[ \\var{2*l*h} + \\var{2*w*h} + \\var{l*w} \\,\\mathrm{m}^2 = \\var{rall}\\,\\mathrm{m}^2 \\text{.} \\]

c)

The exact number of buckets needed is

\\[\\var{rall}\\,\\text{m}^2 \\div 15\\,\\text{m}^2 = \\var{rall/15} \\text{.}\\]

{person['name']} can only buy a whole number of buckets, so {pronouns['they']} need{verbs} to decide between {buckets-1} and {buckets} paint buckets. As it is better to buy more paint than not buy enough, {pronouns['they']} should buy {buckets} buckets of {colour} paint.

"}, {"name": "Converting units of length (km/m/miles)", "extensions": ["random_person"], "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/"}], "variable_groups": [], "variables": {"miles": {"templateType": "anything", "definition": "km*km_to_miles", "description": "", "name": "miles", "group": "Ungrouped variables"}, "location": {"templateType": "anything", "definition": "random('Boston', 'Edinburgh', 'Great Welsh', 'London')", "description": "", "name": "location", "group": "Ungrouped variables"}, "km_to_miles": {"templateType": "anything", "definition": "0.62", "description": "

conversion rate km to miles

", "name": "km_to_miles", "group": "Ungrouped variables"}, "person": {"templateType": "anything", "definition": "random_person()", "description": "", "name": "person", "group": "Ungrouped variables"}, "km": {"templateType": "anything", "definition": "random(24..28#2)", "description": "", "name": "km", "group": "Ungrouped variables"}}, "type": "question", "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "km*1000", "maxValue": "km*1000", "marks": 1, "variableReplacements": []}], "prompt": "

The week before the event, {person['name']} goes on a final training run. An app on {person['pronouns']['their']} phone tells {person['pronouns']['them']} that {person['pronouns']['they']} ran $\\var{km}$ km.

What was the length of {person['pronouns']['their']} training run in metres?

[[0]] metres.

"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "strictPrecision": false, "maxValue": "miles+0.5", "precision": 0, "marks": 1, "mustBeReduced": false, "variableReplacements": [], "minValue": "miles-0.5", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}], "prompt": "

Use the approximate conversion, $1$ kilometre = $0.62$ miles, to find the length of {person['pronouns']['their']} training run in miles.

[[0]] miles. Round your answer to the nearest mile.

"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "42-0.5", "maxValue": "42+0.5", "marks": 1, "variableReplacements": []}], "prompt": "

{person['name']} knows that the {location} Marathon is 26 miles. Use the conversion rate in part b) to find the approximate length in km.

[[0]] km round your answer to the nearest km

"}], "advice": "

a)

{person['name']}'s training run is $\\var{km}$ km long.

To convert $\\var{km}$ km into metres, we multiply $\\var{km}$ by $1000$.

\\[\\var{km}\\times1000= \\var{km*1000}\\text{ metres.}\\]

b)

To convert $\\var{km}$ km into miles, we multiply $\\var{km}$ by the conversion rate given: $0.62$.

\\[\\begin{align}
\\var{km}\\times\\frac{5}{8}&= \\var{miles}\\\\
&=\\var{dpformat(miles,0)}\\text{ miles, rounded to the nearest integer.}
\\end{align}\\]

c)

The {location} Marathon is $26$ miles long. To convert to km we multiply by the inverse of the conversion rate given in part b):

\\[ 26 \\times \\frac{1}{0.62} = 42\\text{ miles, rounded to the nearest integer.} \\]

", "tags": ["taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["km_to_miles", "km", "miles", "person", "location"], "statement": "

{person['name']} is training for the {location} Marathon.

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Convert from km to metres and miles, and miles to km.

"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Converting units of height (feet/inches/cm)", "extensions": ["random_person"], "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/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"stepsPenalty": 0, "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "cm", "maxValue": "cm", "marks": "2", "variableReplacements": []}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

What is {person['name']}'s height in centimetres?

[[0]]cm round your answer to the nearest cm

", "steps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "60", "maxValue": "60", "marks": "0.5", "variableReplacements": [], "prompt": "

We have information on how to convert feet to inches and inches to cm, but not feet to cm. We will therefore first convert the height into inches only.

What is $5$ft in inches?

"}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "60+inches", "maxValue": "60+inches", "marks": "0.5", "variableReplacements": [], "prompt": "

What is {person['name']}'s height in inches?

"}, {"variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

We can now use the conversion rate for inches to cm to find {person['name']}'s height in cm.

"}]}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "mustBeReducedPC": 0, "showFeedbackIcon": true, "precisionType": "dp", "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "strictPrecision": false, "minValue": "cm/100", "maxValue": "cm/100", "marks": 1, "variableReplacements": [], "showPrecisionHint": false, "precision": "2", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

What is {person['name']}'s height in metres?

[[0]]m round your answer to 2 decimal places

"}], "advice": "

{person['name']} is $5$ft $\\var{inches}$ inches tall.

a)

To find {person['name']}'s height into cm, we first convert it into inches.

We can convert feet into inches by multiplying by $12$. So $5$ feet is

\\[ 5\\times12=60 \\text{ inches.}\\]

Therefore $5$ft $\\var{inches}$ inches is

\\[
60+\\var{inches} = \\var{60+inches}\\text{ inches.}
\\]

We can now convert this to cm by multiplying by $2.54$.

\\[\\var{60+inches}\\times2.54=\\var{cm}\\text{ cm, rounded to the nearest integer.}\\]

b)

To convert centimetres into metres, we divide by $100$:

\\[\\var{cm}\\div100=\\var{dpformat(cm/100,2)} \\text{ metres.}\\]

Therefore, {person['name']} is $\\var{dpformat(cm/100,2)}$ metres tall.

", "tags": ["taxonomy"], "variables": {"cm": {"templateType": "anything", "description": "", "definition": "precround((inchfeet*2.54),0)", "name": "cm", "group": "Ungrouped variables"}, "inchfeet": {"templateType": "anything", "description": "", "definition": "(5*12)+inches", "name": "inchfeet", "group": "Ungrouped variables"}, "person": {"templateType": "anything", "description": "", "definition": "random_person()", "name": "person", "group": "Ungrouped variables"}, "s": {"templateType": "anything", "description": "", "definition": "if(person['gender']='neutral','','s')", "name": "s", "group": "Ungrouped variables"}, "inches": {"templateType": "anything", "description": "", "definition": "random(4..11)", "name": "inches", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["person", "inches", "inchfeet", "cm", "s"], "statement": "

{person['name']} is $5$ft $\\var{inches}$ inches tall and would like to find {person['pronouns']['their']} height in cm.

{capitalise(person['pronouns']['they'])} find{s} the following unit conversion table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$1$ foot	$12$ inches
$1$ inch	$2.54$ cm
$1$ metre	$100$ cm

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Convert a height given in feet and inches into cm and then metres.

"}, "variablesTest": {"condition": "", "maxRuns": "1000"}}, {"name": "Using compound units - speed", "extensions": [], "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/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "precisionMessage": "

Round your answer to $2$ decimal places.

", "precisionType": "dp", "mustBeReducedPC": 0, "precisionPartialCredit": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "distance/seconds", "precision": "2", "marks": 1, "scripts": {}, "minValue": "distance/seconds", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

What was the runner's average speed, in metres per second?

[[0]] m/s

"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "precisionMessage": "

Round your answer to $2$ decimal places.

", "precisionType": "dp", "mustBeReducedPC": 0, "precisionPartialCredit": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "distance/seconds*3.6", "precision": "2", "marks": 1, "scripts": {}, "minValue": "distance/seconds*3.6", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

How fast is this in kilometres per hour?

[[0]] km/h

"}], "advice": "

a)

To find the average speed of the runner in meters per second (m/s), we divide the distance covered by the runner (in metres) by the time taken for the runner to run this distance (in seconds).

\\[
\\begin{align}
\\text{Average speed} &= \\displaystyle\\frac{\\var{distance}}{\\var{seconds}}\\\\
&= \\var{distance/seconds}\\\\
&= \\var{dpformat(distance/seconds,2)}\\; \\text{m/s} \\; (\\text{rounded to $2$ decimal places}).
\\end{align}
\\]

b)

We can convert the average speed of the runner that we calculated in a) in metres per second to kilometres per hour using the following two equivalences:

\\[1\\text{m} = \\displaystyle\\frac{1}{1000}\\text{km},\\]

\\[
1 \\; \\text{second} = \\displaystyle\\frac{1}{60} \\; \\text{minutes} = \\displaystyle\\frac{1}{3600} \\; \\text{hours}.
\\]

We know from a) that the average speed of the runner in m/s was $\\var{dpformat(distance/seconds,5)}$ m/s ($5$ d.p), so to convert this speed to km/h we first need to convert metres to kilometres,

\\[\\var{dpformat(distance/seconds,5)} \\; \\text{m/s} = \\var{dpformat(distance/seconds/1000,5)} \\text{km/s} \\; (5 \\; \\text{d.p})\\]

Then we convert seconds to hours,

\\[1 \\; \\text{second} = \\displaystyle\\frac{1}{3600} \\; \\text{hours}.\\]

Now we have

\\[\\var{dpformat(distance/seconds,5)} \\; \\text{m/s} = \\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours}.\\]

We want a rate per one hour, so we multiply by $3600$ to obtain a measurement in km/h:

\\[\\begin{align}\\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours} &= \\var{siground(distance/seconds/1000,5)*3600} \\; \\text{km}/\\text{h}\\\\&=\\var{dpformat(distance/seconds*3.6, 2)} \\; \\text{km}/\\text{h} \\; (\\text{rounded to $2$ decimal places}).\\end{align}\\]

\\[\\var{sigformat(distance/seconds/1000,5)} \\; \\text{kilometres per $\\displaystyle\\frac{1}{3600}$ hours} = \\var{distance/seconds*3.6} \\; \\text{km}/\\text{h}.\\]

Note that throughout this calculation we have rounded all figures to $5$ decimal places for convenience; when doing calculations which involve long decimals, you should always input the full figure into your calculator to avoid getting an incorrect answer due to rounding.

", "tags": ["taxonomy"], "variables": {"speed": {"templateType": "anything", "description": "

Athlete's speed, in m/s.

4m/s is about 9 mph, a bit faster than a jog. The current world record is 12m/s.

", "definition": "random(4..8#0)", "name": "speed", "group": "Ungrouped variables"}, "distance": {"templateType": "anything", "description": "

Distance that the runner ran.

", "definition": "random(80,100,150,200)", "name": "distance", "group": "Ungrouped variables"}, "seconds": {"templateType": "anything", "description": "

Time taken to cover the distance, in seconds.

", "definition": "floor(distance/speed)", "name": "seconds", "group": "Ungrouped variables"}}, "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": "100"}, "functions": {}, "ungrouped_variables": ["distance", "seconds", "speed"], "statement": "

An athlete runs $\\var{distance}$ m in $\\var{seconds}$ seconds.

Round each of your answers to two decimal places.

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Calculate a speed in m/s given distance and time taken, then convert that to km/hour

"}}, {"name": "Using compound units - room hire price per hour and per minute", "extensions": ["random_person"], "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/"}], "tags": ["taxonomy"], "metadata": {"description": "

Given the cost of hiring a room for a given number of hours, compare with competing prices given per hour and per minute.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{pname} has been tasked with booking a room for a {hours}-hour meeting.

", "advice": "

a)

The price per hour is the total price divide by the number of hours.

\\[ \\text{Price per hour} = \\frac{\\var{block_price_per_hour*hours}}{\\var{hours}} = £\\var{dpformat(block_price_per_hour,2)} \\text{ per hour} \\]

b)

The price is given in pence per minute. To convert to pounds per minute, divide by $100$:

\\[ \\var{100*competitor_price_per_minute} \\text{ p/minute} = £\\var{dpformat(competitor_price_per_minute,2)} \\text{ per minute} \\]

Then to convert to pounds per hour, multiply by $60$:

\\[ £\\var{dpformat(competitor_price_per_minute,2)} \\text{ per minute} = £\\var{dpformat(competitor_price_per_minute*60,2)} \\text{ per hour} \\]

c)

{pname} should choose the method with the lowest cost per hour, which is {best_method}.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"competitor_price_per_minute": {"name": "competitor_price_per_minute", "group": "Ungrouped variables", "definition": "floor(100*block_price_per_hour/60*(1+random(0.1..0.3#0)*random(-1,1)))/100", "description": "

Price of booking at RoomCo, the competitor, in pounds per minute

", "templateType": "anything", "can_override": false}, "pronouns": {"name": "pronouns", "group": "Person", "definition": "person['pronouns']", "description": "", "templateType": "anything", "can_override": false}, "best_method": {"name": "best_method", "group": "Ungrouped variables", "definition": "switch(\n min(prices)=block_price_per_hour,\n 'paying in advance at ACME',\n min(prices)=single_price_per_hour,\n 'pay-as-you-go at ACME',\n 'paying per minute at RoomCo'\n)", "description": "

A description of the cheapest method.

", "templateType": "anything", "can_override": false}, "block_price_per_hour": {"name": "block_price_per_hour", "group": "Ungrouped variables", "definition": "random(10..25#0.25)", "description": "

Price of booking the room at ACME in advance, in pounds per hour

", "templateType": "anything", "can_override": false}, "hours": {"name": "hours", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "

Length of the meeting in hours

", "templateType": "anything", "can_override": false}, "pname": {"name": "pname", "group": "Person", "definition": "person['name']", "description": "", "templateType": "anything", "can_override": false}, "marking_matrix": {"name": "marking_matrix", "group": "Ungrouped variables", "definition": "let(best,min(prices),\n map(if(x=best,1,0),x,prices)\n)", "description": "

Marking matrix for the \"which method is best\" part.

", "templateType": "anything", "can_override": false}, "single_price_per_hour": {"name": "single_price_per_hour", "group": "Ungrouped variables", "definition": "block_price_per_hour+random(0.5..2#0.25)*random(-1,1)", "description": "

Pay-as-you-go price at ACME, in pounds per hour

", "templateType": "anything", "can_override": false}, "verbs": {"name": "verbs", "group": "Person", "definition": "if(person['gender']='neutral','','s')", "description": "", "templateType": "anything", "can_override": false}, "prices": {"name": "prices", "group": "Ungrouped variables", "definition": "[block_price_per_hour,single_price_per_hour,60*competitor_price_per_minute]", "description": "", "templateType": "anything", "can_override": false}, "person": {"name": "person", "group": "Person", "definition": "random_person()", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["hours", "block_price_per_hour", "single_price_per_hour", "competitor_price_per_minute", "marking_matrix", "prices", "best_method"], "variable_groups": [{"name": "Person", "variables": ["person", "pronouns", "pname", "verbs"]}], "functions": {"pounds": {"parameters": [["n", "number"]], "type": "string", "language": "jme", "definition": "currency(n,\"\u00a3\",\"p\")"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

{pname} is quoted a price of {pounds(block_price_per_hour*hours)} by ACME Office Services to book a room in advance for {hours} hours, or {pounds(single_price_per_hour)} per hour in a pay-as-you-go scheme.

To compare the two prices, {pronouns['they']} decide{verbs} to convert the advance booking price to a price per hour.

Price per hour: £ [[0]]

", "gaps": [{"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, "minValue": "block_price_per_hour", "maxValue": "block_price_per_hour", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

A competitor, RoomCo, is offering meeting rooms charged by the minute, at {pounds(competitor_price_per_minute)} per minute.

To compare this price to ACME's offer, {pname} decide{verbs} to convert it to a price per hour.

Price per hour: £ [[0]]

", "gaps": [{"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, "minValue": "60*competitor_price_per_minute", "maxValue": "60*competitor_price_per_minute", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

How should {pname} book the room?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["

Pay in advance at ACME

", "

Pay-as-you-go at ACME

", "

Pay per minute at RoomCo

"], "matrix": "marking_matrix"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Calculate the areas of polygons", "extensions": [], "custom_part_types": [], "resources": [["question-resources/trapezium.svg", "/srv/numbas/media/question-resources/trapezium.svg"], ["question-resources/trangle.svg", "/srv/numbas/media/question-resources/trangle.svg"], ["question-resources/parallelogram.svg", "/srv/numbas/media/question-resources/parallelogram.svg"], ["question-resources/Parallelogram_area_animated.gif", "/srv/numbas/media/question-resources/Parallelogram_area_animated.gif"], ["question-resources/rectangle_zISmvoz.svg", "/srv/numbas/media/question-resources/rectangle_zISmvoz.svg"], ["question-resources/hardertrapezium_8GqMwOo.svg", "/srv/numbas/media/question-resources/hardertrapezium_8GqMwOo.svg"], ["question-resources/Trap_advice.svg", "/srv/numbas/media/question-resources/Trap_advice.svg"], ["question-resources/Triangle_advice_lD6eKvD.svg", "/srv/numbas/media/question-resources/Triangle_advice_lD6eKvD.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "rulesets": {}, "functions": {}, "ungrouped_variables": [], "metadata": {"description": "

This question tests the students ability to calculate the area of different 2D shapes given the units and measurements required. The formulae for the areas are available if required but students are encouraged to try to remember them themselves.

The shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.

Author of gif: Picknick
https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [{"variables": ["h1", "w1", "wh11", "wh11dp"], "name": "Parallelogram"}, {"variables": ["h2", "w2", "wh22", "wh22dp"], "name": "Triangle"}, {"variables": ["h5", "w5a", "w5b", "wabh5dp", "wabh5"], "name": "'Harder' trapezium"}, {"variables": ["w0", "h0", "wh00", "wh00dp"], "name": "Rectangle"}], "advice": "

The area of a rectangle is calculated using the formula

\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}\\text{.}\\]

We have a base of $\\var{w0}$m and a height $\\var{h0}$m, therefore

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\ &= \\var{w0*h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

The parallelogram is just a slanted rectangle:

$\"Parallelogram\"$

Animation by Picknick.

Therefore, the area of a parallelogram is calculated using the formula

\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]

We have a base $\\var{w1}$m and perpendicular height $\\var{h1}$m.

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\ &= \\var{{w1}{h1}}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

c)

The area of a triangle is calculated using the formula

\\[\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\\]

Note that the triangle is half of a rectangle:

Our triangle has a base $\\var{w2}$m and a height $\\var{h2}$m, therefore

\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{0.5*w2*h2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}

A trapezium can be interpreted as half of a parallelogram, this is shown below:

As we only want the area of one half of this shape, the area is half of

\\[\\mathrm{area} = (a+b) \\times \\mathrm{height}\\text{,}\\]

with ${a} = \\var{w5a}$m, ${b} = \\var{w5b}$m, and height $\\var{h5}$m.

\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*(h5)/2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}

\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}

", "statement": "

Calculate the area of the following shapes.

", "preamble": {"css": "", "js": ""}, "tags": ["area", "Area", "area of a parallelogram", "area of a rectangle", "area of a right-angled triangle", "area of a trapezium", "parallelogram", "Rectangle", "rectangle", "right - angled triangle", "shapes", "taxonomy", "trapezium"], "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The area of the rectangle is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "{h0}{w0}", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "{h0}{w0}", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a rectangle is:

\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "prompt": "

The area of the parallelogram is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "{h1}*{w1} + 0.01", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "{h1}*{w1} - 0.01", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a parallelogram is:

\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]

The area of the triangle is [[0]] $\\mathrm{m^2}$ Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "{w2}{h2}*0.5 + 0.01", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "{w2}{h2}*0.5 - 0.01", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a triangle is:

\\[\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\\]

The area of the trapezium is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.

", "gaps": [{"variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "maxValue": "0.5{w5a+w5b}{h5} + 0.01", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "minValue": "0.5{w5a+w5b}{h5} - 0.01", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "marks": "2", "scripts": {}}], "stepsPenalty": "1", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "

The formula for the area of a trapezium is:

\\[\\mathrm{Area} = \\frac{(a+b)}{2}\\times \\mathrm{height}.\\]

", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "variables": {"h2": {"name": "h2", "description": "

Height of the triangle.

", "templateType": "anything", "group": "Triangle", "definition": "random(1..4.5#0.1)"}, "wh22dp": {"name": "wh22dp", "description": "

The Area of a triangle using the two terms, w2 and h2 to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "Triangle", "definition": "precround(0.5*w2*h2, 1)"}, "wabh5": {"name": "wabh5", "description": "

The Area of a trapezium using the three terms, w5a, w5b and h5, such that a condition can be satisfied.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "precround((w5a+w5b)*(h5)/2, 5)"}, "w5b": {"name": "w5b", "description": "

The bottom parallel side in the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(7.5..10#0.1)"}, "wh11": {"name": "wh11", "description": "

The product of the two terms, w1 and h1, such that a condition can be satisfied.

", "templateType": "anything", "group": "Parallelogram", "definition": "precround(w1*h1,3)"}, "w1": {"name": "w1", "description": "

The width of the parallelogram.

", "templateType": "anything", "group": "Parallelogram", "definition": "random(5..10#0.1)"}, "wabh5dp": {"name": "wabh5dp", "description": "

The Area of a trapezium using the three terms, w5a, w5b and h5 to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "precround((w5a+w5b)*(h5)/2, 1)"}, "h1": {"name": "h1", "description": "

The height of the parallelogram

", "templateType": "anything", "group": "Parallelogram", "definition": "random(1..4.5#0.1)"}, "w0": {"name": "w0", "description": "

Width of the rectangle.

", "templateType": "anything", "group": "Rectangle", "definition": "random(5..10#0.1)"}, "wh22": {"name": "wh22", "description": "

The Area of a triangle using the two terms, w2 and h2, such that a condition can be satisfied.

", "templateType": "anything", "group": "Triangle", "definition": "precround(0.5*w2*h2,4)"}, "w2": {"name": "w2", "description": "

Base of the triangle.

", "templateType": "anything", "group": "Triangle", "definition": "random(5..10#0.1)"}, "h5": {"name": "h5", "description": "

Height of the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(2..5#0.1)"}, "wh00dp": {"name": "wh00dp", "description": "

The product of the two terms, w0 and h0, to one decimal place, such that a condition can be satisfied.

", "templateType": "anything", "group": "Rectangle", "definition": "precround(w0*h0,1)"}, "wh00": {"name": "wh00", "description": "

The product of the two terms, w0 and h0, such that a condition can be satisfied.

", "templateType": "anything", "group": "Rectangle", "definition": "precround(w0*h0,3)"}, "h0": {"name": "h0", "description": "

Height of the rectangle.

", "templateType": "anything", "group": "Rectangle", "definition": "random(1..5#0.1)"}, "w5a": {"name": "w5a", "description": "

The top parallel side in the trapezium.

", "templateType": "anything", "group": "'Harder' trapezium", "definition": "random(5..6.5#0.1)"}, "wh11dp": {"name": "wh11dp", "description": "

The product of the two terms, w1 and h1, to one decimal place such that a condition can be satisfied.

", "templateType": "anything", "group": "Parallelogram", "definition": "precround(w1*h1, 1)"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Using compound units: price/weight of sweets", "extensions": ["random_person"], "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": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "tags": ["Compound units", "compound units", "conversion", "measurements", "rate of pay", "speed", "taxonomy", "unit pricing", "using compound units"], "metadata": {"description": "

This question assesses the students ability to calculate and convert between different types of compound units, including rates of pay, speed and unit pricing.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{pname} goes to {pronouns['their']} local shop and buys a bag containing $\\var{weight}$g of sweets for £$\\var{cost}$.

", "advice": "

a)

We are given the price of a bag of $\\var{weight}$ grams of sweets.

To find the price per 100g of sweets we divide the price of a bag of sweets by its weight in grams and then multiply this by $100$.

\\[\\displaystyle\\frac{\\var{cost}}{\\var{weight}} \\times 100 = \\var{(100*cost/weight)}.\\]

\\begin{align}
\\displaystyle\\frac{\\var{cost}}{\\var{weight}} \\times 100 &= \\var{(100*cost/weight)}\\\\ &= \\var{dpformat(100*cost/weight,2)} \\; (\\text{rounded to $2$ decimal places}).
\\end{align}

The sweets cost {pounds(100*cost/weight)} per 100g.

b)

To convert the cost from pounds per $100$ grams to pounds per kilogram we need to use the fact that $1\\text{g} = \\displaystyle\\frac{1}{1000}\\text{kg}$.

This means that $100\\text{g} = \\displaystyle\\frac{1}{10}\\text{kg}$.

We know from a) that sweets cost {pounds(100*cost/weight)} per 100g, which is the same as {pounds(100*cost/weight)} per $\\frac{1}{10}$kg.

We want the price per one kilogram of sweets, so we multiply by $10$.

Note that we use the actual value of $\\displaystyle\\frac{\\var{cost}}{\\var{weight}} \\times 100 = \\var{100*cost/weight}$ here to ensure that our final answer is accurate.

\\begin{align}
\\var{100*cost/{weight}} \\times 10 &= \\var{dpformat(1000*{cost}/{weight},2)} \\; (2 \\; \\text{d.p})
\\end{align}

So, the sweets cost {pounds(1000*cost/weight)} per kg.

c)

We worked out in part a) that sweets cost {pounds(100*cost/weight)} per 100g when bought in the bag, so at {pounds(pick_n_mix_cost)} per 100g the Pick'n'Mix is {if(pick_n_mix_cost<100*cost/weight,'cheaper','more expensive')} than buying the bag.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pronouns": {"name": "pronouns", "group": "Ungrouped variables", "definition": "person['pronouns']", "description": "", "templateType": "anything", "can_override": false}, "cost": {"name": "cost", "group": "Ungrouped variables", "definition": "precround(cost_per_g*weight,1)-0.01", "description": "

Cost of a bag of sweets - always ends in .x9 to look like a real price.

", "templateType": "anything", "can_override": false}, "max_kg_cost": {"name": "max_kg_cost", "group": "Ungrouped variables", "definition": "max(precround(1000*cost/weight,2), precround(100*cost/weight,2)*10)", "description": "

Minimum acceptable cost per kg - using the rounded cost per 100g can introduce an error.

", "templateType": "anything", "can_override": false}, "pname": {"name": "pname", "group": "Ungrouped variables", "definition": "person['name']", "description": "", "templateType": "anything", "can_override": false}, "pick_n_mix_cost": {"name": "pick_n_mix_cost", "group": "Ungrouped variables", "definition": "precround(100*cost/weight,2)+random(-15..15 except -2..2)*0.01", "description": "

Cost of the sweets at the Pick'n'Mix, per 100g.

", "templateType": "anything", "can_override": false}, "min_kg_cost": {"name": "min_kg_cost", "group": "Ungrouped variables", "definition": "min(precround(1000*cost/weight,2), precround(100*cost/weight,2)*10)", "description": "

Minimum acceptable cost per kg - using the rounded cost per 100g can introduce an error.

", "templateType": "anything", "can_override": false}, "weight": {"name": "weight", "group": "Ungrouped variables", "definition": "random(150..200)", "description": "

Weight of a bag of sweets, in grams

", "templateType": "anything", "can_override": false}, "person": {"name": "person", "group": "Ungrouped variables", "definition": "random_person()", "description": "", "templateType": "anything", "can_override": false}, "cost_per_g": {"name": "cost_per_g", "group": "Ungrouped variables", "definition": "random(0.005..0.02#0)", "description": "

Cost of the sweets per gram, in pounds.

Between 50p and £2 per 100g.

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "10000"}, "ungrouped_variables": ["weight", "cost_per_g", "cost", "min_kg_cost", "max_kg_cost", "person", "pname", "pronouns", "pick_n_mix_cost"], "variable_groups": [], "functions": {"pounds": {"parameters": [["n", "number"]], "type": "string", "language": "jme", "definition": "currency(n,\"\u00a3\",\"p\")"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

How much do the sweets cost per $100$ grams?

£[[0]] per $100$g

Round your answer to $2$ decimal places.

", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

How much is this in pounds per kilogram?

£[[0]] per kg

Round your answer to $2$ decimal places.

", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

{pname} notices that the same sweets are available from the Pick'n'Mix for {pounds(pick_n_mix_cost)} per 100g.

Should {pronouns['they']} buy {pronouns['their']} sweets from the Pick'n'Mix instead?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

Yes, the Pick'n'Mix is cheaper.

", "

No, the Pick'n'Mix is more expensive.

"], "matrix": "if(pick_n_mix_cost<100*cost/weight,[1,0],[0,1])"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Converting units: baby weight", "extensions": ["stats"], "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/"}], "variable_groups": [{"variables": ["oz_to_kg", "weight_kg", "weight_lb", "weight_oz", "weight_oz_rem"], "name": "hospital baby"}, {"variables": ["news_reports", "news", "newskg"], "name": "News baby"}], "preamble": {"css": "", "js": ""}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "weight_lb", "maxValue": "weight_lb", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "weight_oz_rem", "maxValue": "weight_oz_rem", "marks": 1, "variableReplacements": []}], "marks": 0, "variableReplacements": [], "prompt": "

A hospital measures the weight of a baby to be $\\var{weight_kg}$kg, but they are asked for the weight in pounds and ounces by the baby's parents, who would like to make an announcement to their friends and family in the traditional way.

Using the conversion table above, what is the weight in pounds (lb) and ounces (oz)?

[[0]]lb and [[1]]oz Round your answer to the nearest ounce.

", "steps": [{"correctAnswerFraction": false, "precisionType": "dp", "showCorrectAnswer": true, "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showPrecisionHint": true, "minValue": "weight_oz", "maxValue": "weight_oz", "variableReplacements": [], "prompt": "

Using the conversion rate for kg to oz, what is the weight of the baby in ounces?

", "precisionMessage": "You have not given your answer to the correct precision.", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "precision": 0, "marks": "0.5", "mustBeReduced": false}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "weight_lb", "maxValue": "weight_lb", "marks": "0.5", "variableReplacements": [], "prompt": "

Now use the conversion rate for lb to oz. Rounding down, how many whole pounds does the baby weigh?

"}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "weight_lb*16", "maxValue": "weight_lb*16", "marks": "0.5", "variableReplacements": [], "prompt": "

What is the above weight in ounces?

"}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "weight_oz_rem", "maxValue": "weight_oz_rem", "marks": "0.5", "variableReplacements": [], "prompt": "

Subtract the above weight from the baby's weight in ounces found in the first step. What is the remainder in ounces?

"}], "showFeedbackIcon": true, "scripts": {}, "stepsPenalty": 0}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "mustBeReducedPC": 0, "precisionPartialCredit": 0, "showFeedbackIcon": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "minValue": "newskg", "maxValue": "newskg", "precision": "1", "marks": 1, "variableReplacements": [], "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision."}], "marks": 0, "variableReplacements": [], "prompt": "

In {news['year']}, newspapers reported the birth of {news['text']} with a weight over $1$ stone. The baby was recorded at $1$ stone, $\\var{news['lb']}$ pounds and $\\var{news['oz']}$ ounces.

What was the weight in kg?

[[0]] kg round your answer to 1 decimal place.

", "scripts": {}, "showFeedbackIcon": true}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Convert weights of babies between kg and lb and oz.

"}, "tags": ["taxonomy"], "variables": {"weight_oz_rem": {"templateType": "anything", "definition": "mod(weight_oz,16)", "description": "

The remainder in ounces

", "name": "weight_oz_rem", "group": "hospital baby"}, "weight_lb": {"templateType": "anything", "definition": "floor(weight_oz/16)", "description": "

Weight of the baby in lb

", "name": "weight_lb", "group": "hospital baby"}, "news_reports": {"templateType": "json", "definition": "json_decode(safe(\"[\\n {\\n \\\"year\\\": 2016,\\n \\\"text\\\": \\\"only the second baby to be born in the UK\\\",\\n \\\"lb\\\": 1,\\n \\\"oz\\\": 8\\n },\\n {\\n \\\"year\\\": 2014,\\n \\\"text\\\": \\\"a boy in California\\\",\\n \\\"lb\\\": 2,\\n \\\"oz\\\": 2\\n },\\n {\\n \\\"year\\\": 2014,\\n \\\"text\\\": \\\"a boy in Florida, the heaviest ever recorded in the state,\\\",\\n \\\"lb\\\": 0,\\n \\\"oz\\\": 2\\n }, \\n {\\n \\\"year\\\": 2012,\\n \\\"text\\\": \\\"a record-breaking baby in China,\\\",\\n \\\"lb\\\": 1,\\n \\\"oz\\\": 5\\n } \\n]\"))", "description": "

A list of news reports

", "name": "news_reports", "group": "News baby"}, "newskg": {"templateType": "anything", "definition": "((16*(14+news['lb']))+news['oz'])/oz_to_kg", "description": "

Weight in kg of the baby in the news

", "name": "newskg", "group": "News baby"}, "news": {"templateType": "anything", "definition": "random(news_reports)", "description": "

The news report we're using

", "name": "news", "group": "News baby"}, "oz_to_kg": {"templateType": "anything", "definition": "35.3", "description": "

conversion rate, ounces in a kg

", "name": "oz_to_kg", "group": "hospital baby"}, "weight_kg": {"templateType": "anything", "definition": "precround(\n min(max(normalsample(3.5,1/1.96),2.5),5)\n,1)", "description": "

Weight of the baby in kg, given to the student in the question.

Wikipedia says that the average baby of European heritage weighs 3.5kg, with 90% between 2.5 and 5.

The min and max are to avoid really odd-looking weights.

", "name": "weight_kg", "group": "hospital baby"}, "weight_oz": {"templateType": "anything", "definition": "precround(weight_kg*oz_to_kg,0)", "description": "

weight of the baby in oz, rounded to the nearest integer

", "name": "weight_oz", "group": "hospital baby"}}, "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "ungrouped_variables": [], "statement": "

In many countries, it is still traditional to announce the weight of a baby in imperial units (pounds and ounces), rather than metric (grams or kilograms).

A conversion table is as follows:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$1$ pound (lb)	$16$ ounces (oz)
$1$ stone (st)	$14$ pounds (lb)
$1$ kg	$\\var{oz_to_kg}$ ounces (oz)

", "advice": "

a)

We are told that the baby's weight is $\\var{weight_kg}$kg and we would like to convert this to pounds and ounces.

To do this, we first obtain the weight in ounces using the conversion given:

\\begin{align}
\\var{weight_kg} \\text{ kg} &= \\var{weight_kg} \\times \\var{oz_to_kg} \\text{ oz} \\\\
&= \\var{weight_oz}\\text{ oz.}
\\end{align}

We can now divide by $16$ to find out how many whole pounds we have:

\\[ \\var{weight_oz}\\text{ oz} \\div 16 = \\var{dpformat(weight_oz/16,1)}\\text{ lb.} \\]

There are $\\var{weight_lb}$ whole pounds, which leaves a remainder

\\[ \\var{weight_oz} - (16 \\times \\var{weight_lb}) = \\var{weight_oz_rem} \\text{ oz.} \\]

The baby weighs $\\var{weight_lb}$ pounds and $\\var{weight_oz_rem}$ ounces.

b)

We are given the weight of the baby in stones, pounds and ounces. To convert to kg we first convert the baby's weight to ounces.

$1$ stone is $14$ pounds, so the baby weighs $\\var{news['lb']+14}$ pounds and $\\var{news['oz']}$ ounces, or

\\[ (\\var{news['lb']+14} \\times 16) + \\var{news['oz']} = \\var{(news['lb']+14)*16+news['oz']} \\text{ oz.}\\]

We can now convert to kg:

\\begin{align}
\\var{(news['lb']+14)*16+news['oz']} \\text{ oz} &= \\var{(news['lb']+14)*16+news['oz']} \\div \\var{oz_to_kg} \\\\
&= \\var{dpformat(newskg,1)}\\text{ kg.}
\\end{align}

"}, {"name": "Converting units of volume (cc/cm^3/litres/m^3)", "extensions": ["random_person"], "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/"}], "variable_groups": [], "variables": {"metres_cubed": {"templateType": "anything", "definition": "cc*(10^-6)", "description": "", "name": "metres_cubed", "group": "Ungrouped variables"}, "cc": {"templateType": "anything", "definition": "random(1200..3000#200)", "description": "", "name": "cc", "group": "Ungrouped variables"}, "s": {"templateType": "anything", "definition": "if(person['gender']='neutral','','s')", "description": "", "name": "s", "group": "Ungrouped variables"}, "person": {"templateType": "anything", "definition": "random_person()", "description": "", "name": "person", "group": "Ungrouped variables"}, "litres": {"templateType": "anything", "definition": "cc/1000", "description": "", "name": "litres", "group": "Ungrouped variables"}}, "type": "question", "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "litres", "maxValue": "litres", "marks": 1, "variableReplacements": []}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

{person['name']} applies to find out how much the insurance for the car would cost, but is required to state the engine size in litres.

What is the engine size in litres?

[[0]] litres.

"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "mustBeReducedPC": 0, "scripts": {}, "showFeedbackIcon": true, "precisionType": "dp", "minValue": "metres_cubed", "correctAnswerStyle": "plain", "allowFractions": false, "showPrecisionHint": false, "strictPrecision": false, "maxValue": "metres_cubed", "precision": "4", "marks": "2", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}], "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "

The specification of a second car gives the engine size in m$^3$. In order for {person['name']} to make a comparison {person['pronouns']['they']} convert{s} the engine size of the first car to cubic metres.

What is the engine size of the first car in units of m$^3$?

[[0]]m$^3$ Give your answer to 4 decimal places

"}], "advice": "

a)

The advertised engine size is $\\var{cc}$ cubic centimetres. To convert cubic centimetres to litres, we divide by $1000$.

\\[\\var{cc}\\div 1000= \\var{litres}\\text{ litres.}\\]

b)

In order to convert to cubic metres, we first note that

\\[ 1 \\text{cm} = 0.01 \\text{m.} \\]

An example of a volume of $1\\text{cm}^3$ is a cube with $1$cm sides. Converting each side into metres,

\\begin{align}
1\\text{cm}^3 &= 1\\text{cm}\\times1\\text{cm}\\times1\\text{cm} \\\\
&= 0.01\\text{m}\\times0.01\\text{m}\\times0.01\\text{m} \\\\
&= 0.000001\\text{m}^3 \\text{.}
\\end{align}

Therefore $\\var{cc}\\text{cm}^3$ is

\\[ \\var{cc} \\times 0.000001 = \\var{metres_cubed}\\text{m}^3\\text{.} \\]

", "tags": ["taxonomy"], "preamble": {"css": "", "js": ""}, "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "ungrouped_variables": ["person", "litres", "cc", "s", "metres_cubed"], "statement": "

{person['name']} is looking to buy a car. {capitalise(person['pronouns']['they'])} find{s} one advertised with an engine size of $\\var{cc}$cc.

{person['name']} recognises that 'cc' stands for units of cubic centimetres (cm$^3$) and knows the following conversions:

\n\n\n\n\n\n\n\n\n\n\n\n

$1$ m	$100$ cm
$1$ litre	$1000\\text{cm}^3$

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Convert figures for car engine sizes between cc (cm^3), litres, and m^3.

"}}, {"name": "Compound units : shopping for bananas", "extensions": ["random_person"], "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": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "metadata": {"description": "

This question assesses the student's ability to use some given information involving two different units of measurement to rewrite the information as a compound measure.

", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": ["person", "a", "b", "b_pack", "b_single", "mark_matrix"], "advice": "

a)

The pack price at Fine Fare supermarket is {b_pack}, which converted to pence is {b}p.

The price per banana at Fine Fare supermarket is

\\[ \\var{b} \\div 5 = \\var{b_single}\\text{p.}\\]

b)

As the price at Hintons supermarket is cheaper, {person['pronouns']['they']} should shop there.

c)

The new price per banana at Fine Fare Supermarket is

\\begin{align}
\\var{b} \\div 6 &= \\var{if(isint(b/6),b/6,dpformat(b/6,1))}\\text{p.} \\\\
\\end{align}

Since the price at Hintons Supermarket is still cheaper, {person['name']} should not change {person['pronouns']['their']} decision.

Since the price at Fine Fare Supermarket is now cheaper, {person['name']} should change {person['pronouns']['their']} decision.

", "variable_groups": [], "statement": "

{person['name']} is shopping for bananas and has two local supermarkets:

Hintons Supermarket charges {a}p per banana.

Fine Fare Supermarket charges {b_pack} for a pack of 5 bananas.

", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "

What is the price per banana (in pence) at Fine Fare Supermarket?

[[0]]p/banana

", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "b_single", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "b_single", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

If {person['name']} is interested in getting the best value for money per banana, which supermarket should {person['pronouns']['they']} shop at?

", "choices": ["

Hintons Supermarket

", "

Fine Fare Supermarket

"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "maxMarks": 0, "marks": 0, "distractors": ["", ""], "matrix": ["1", 0], "displayType": "radiogroup"}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "prompt": "

{person['name']} notices that some of the bags of bananas at Fine Fare Supermarket contain 6 bananas and wonders if that will make a difference to {person['pronouns']['their']} decision. Assuming that {person['pronouns']['they']} can get hold of a pack of 6 bananas at Fine Fare, which supermarket should {person['pronouns']['they']} now shop at?

", "choices": ["

Hintons Supermarket

", "

Fine Fare Supermarket

"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": "mark_matrix", "displayType": "radiogroup"}], "tags": ["compound measures", "Compound measures", "Compound units of measurement", "price", "rate of pay", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"b_pack": {"description": "

formatted price of the pack

", "group": "Ungrouped variables", "definition": "currency(b/100,\"\u00a3\",\"p\")", "name": "b_pack", "templateType": "anything"}, "b_single": {"description": "

Single banana price at fine fare

", "group": "Ungrouped variables", "definition": "b/5", "name": "b_single", "templateType": "anything"}, "a": {"description": "

cost per banana in Hintons

", "group": "Ungrouped variables", "definition": "random(12..19)", "name": "a", "templateType": "anything"}, "b": {"description": "

Price of pack of bananas at Fine Fare

", "group": "Ungrouped variables", "definition": "a*5+random(5..20#5 except a)", "name": "b", "templateType": "anything"}, "person": {"description": "", "group": "Ungrouped variables", "definition": "random_person()", "name": "person", "templateType": "anything"}, "mark_matrix": {"description": "", "group": "Ungrouped variables", "definition": "if(b/6>a,[1,0],[0,1])", "name": "mark_matrix", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "showstudentname": true, "percentPass": 0, "type": "exam", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}], "extensions": ["geogebra", "random_person", "stats"], "custom_part_types": [], "resources": [["question-resources/icecrea_QqVaCIf.svg", "/srv/numbas/media/question-resources/icecrea_QqVaCIf.svg"], ["question-resources/frisbee_variable_TESZa4J.svg", "/srv/numbas/media/question-resources/frisbee_variable_TESZa4J.svg"], ["question-resources/tennis-ball_with_variable_MBOLQeM.svg", "/srv/numbas/media/question-resources/tennis-ball_with_variable_MBOLQeM.svg"], ["question-resources/trapezium.svg", "/srv/numbas/media/question-resources/trapezium.svg"], ["question-resources/trangle.svg", "/srv/numbas/media/question-resources/trangle.svg"], ["question-resources/parallelogram.svg", "/srv/numbas/media/question-resources/parallelogram.svg"], ["question-resources/Parallelogram_area_animated.gif", "/srv/numbas/media/question-resources/Parallelogram_area_animated.gif"], ["question-resources/rectangle_zISmvoz.svg", "/srv/numbas/media/question-resources/rectangle_zISmvoz.svg"], ["question-resources/hardertrapezium_8GqMwOo.svg", "/srv/numbas/media/question-resources/hardertrapezium_8GqMwOo.svg"], ["question-resources/Trap_advice.svg", "/srv/numbas/media/question-resources/Trap_advice.svg"], ["question-resources/Triangle_advice_lD6eKvD.svg", "/srv/numbas/media/question-resources/Triangle_advice_lD6eKvD.svg"]]}