![\"Parallelogram\"](\"https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif\")
This is a Mathematics ability assessment for entry into the Bachelor of Engineering and Technology course at Chisholm Institute. The entire platform has bee developed by Numbas. There are 32 math questions and 6 instruction question regarding how to put answer in the answe rbox. Please practice the 'how to put answer' section first before starting to answer math questions.
\nYou can seek help but it will reduce your marks.
\nGood luck!
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 7200, "percentPass": "40", "showQuestionGroupNames": true, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Answering Instruction", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "How to enter algebraic expressions - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "d", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..9)", "name": "c", "description": ""}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s", "description": ""}}, "ungrouped_variables": ["a", "s", "b", "c", "d"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "{a}*x^{b}+{c}x+{d}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Inputting polynomials such as $3x^2+5x-2$ is easy : just input 3*x^2+5*x-2.
Try this:
\nInput this polynomial: $\\simplify[all]{{a}*x^{b}+{c}*x+{d}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^2+{a+c}x*y+{a*c}y^2", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": true, "expectedvariablenames": ["x", "y"], "notallowed": {"showStrings": false, "message": "Do not include brackets in your answer.
", "strings": ["("], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Now consider this problem.
\nExpand the brackets and input the resulting expression:
\n$\\simplify[all]{(x+{a}y)(x+{c}y)}=\\;$[[0]]
\nMake sure that you input an expression in your answer such as $xy$ as x*y.
(Do not include brackets in your answer.)
", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example, we look at how you enter algebraic expressions - those involving symbols.
\nThe box next to your input shows what you've written in mathematical notation and is very important as you can check it against the expression you had in mind.
", "tags": ["algebraic expressions", "checked2015", "input", "introduction", "notation", "Numbas", "numbas", "polynomials", "symbols"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Inputting algebraic expressions into Numbas.
"}, "advice": ""}, {"name": "How to enter algebraic fractions - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..15#2)", "name": "b", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except [round(b*c/a),c])", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..16#2)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d"], "functions": {}, "parts": [{"prompt": "Examples
\nSuppose we wanted to input the expression $\\displaystyle \\frac{\\var{a}+\\var{b}x}{\\var{c}+\\var{d}y}$ into the system.
\nWhich of the following input expressions are incorrect?
\n[[0]]
\nChoose the incorrect input(s): (You lose 3 marks if you choose the wrong one!)
\nIf you click on Submit part, then on Show feedback, you will be given more detail on your choices.
\nYou can click on Reveal at the top of the window to see solutions, but it's best to work these through yourself. Remember you can always redo the question by clicking on Try another question like this one at the bottom.
", "scripts": {}, "gaps": [{"displayType": "checkbox", "choices": ["({a}+{b}x)/({c}+{d}y)
{a}+{b}x/({c}+{d}y)
{a}+{b}x/{c}+{d}y
({a}+{b}x)/{c}+{d}y
Input the expression $\\displaystyle \\frac{\\var{b}+\\var{a}y}{\\var{d}+\\var{c}z}$ here: [[0]]
", "scripts": {}, "gaps": [{"answer": "({b}+{a}y)/({d}+{c}z)", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}, {"prompt": "Input the expression $\\displaystyle \\frac {\\var{d} z + \\var{b}} {(x + \\var{a}) (y + \\var{c})}$ here: [[0]]
", "scripts": {}, "gaps": [{"answer": "({d} * z + {b}) / ((x + {a}) * (y + {c}))", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}, {"prompt": "Input the expression $\\displaystyle \\simplify[std]{({a} -(({b} * x + {c}) * e ^ (( -{2}) * x))) / ((x + {2 * b}) * (y -{3* d}))}$ here: [[0]]
", "scripts": {}, "gaps": [{"answer": "({a} -(({b} * x + {c}) * e ^ (( -{2}) * x))) / ((x + {2 b}) * (y -{3* d}))", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.0001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Ratios of Algebraic Expressions.
\nBy this we mean expressions of the form $\\displaystyle \\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are algebraic expressions.
\nIf you want to input such an expression into the system you HAVE TO BE CAREFUL AND USE BRACKETS otherwise mistakes will occur.
\nOnce again, the box displaying your input in mathematical notation beside the input boxes in parts 2, 3 and 4 is very useful as it shows what the system thinks you have entered.
\nFor complicated expressions this is essential as you can check you have written what you really meant.
", "tags": ["algebraic input", "brackets", "checked2015", "input", "introduction", "mathematical expressions", "numbas", "Numbas", "ratios", "Ratios"], "rulesets": {"std": ["all", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Inputting ratios of algebraic expressions.
"}, "advice": "a) The correct input is ({a}+{b}x)/({c}+{d}y) - the rest are incorrect and you should have ticked those.
b) A correct input is ({b} + {a}y) / ({c} + {d}z). Also correct is ({a}y+{b}) / ({c} + {d}z) etc.
c) A correct input is ({d}z + {b}) / ((x + {a})*(y + {c})).
Note the denominator (the bottom of the ratio) has to have two brackets, i.e. ((x + {a})*(y + {c})) as otherwise the expression ({d}z + {b}) / (x + {a})*(y + {c}) is seen by the system as $\\displaystyle \\left(\\simplify[std]{({d} * z + {b}) / (x + {a})}\\right) (y + \\var{c})$
d) A correct input is ({a} -({b}x + {c})*e ^ ( -{2}x)) / ((x + {2*b})*(y -{3*d}))
Input:
\nInput:
\nInput:
\nFUNCTIONS
\nsin(x) not sinx, ln(a) not lna.abs(a).Here are some examples for you to try:
\n(If you want help, press Reveal Answers to see correct inputs in the Advice section.)
", "tags": ["arctan", "brackets", "checked2015", "functions", "input", "introduction", "Numbas", "numbas", "standard functions"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Dealing with functions in Numbas.
"}, "advice": "Correct inputs for these questions are as follows, although there may be other correct ways of inputting these:
\nsin(cos({a}x)+{b})cos(sin({a}x + {b}))abs((x + {c}) / (x + {d}))ln(abs((x + {a}) / (x + {d}))){a}t^({-b})*e^({-c}t)*sin({b}t) + (t + {d}t ^ 3)*e ^ ({c}t)arctan(({c}y ^ 2 + {d}) / ((y + {a})*(y + {b})))Find the result of this calculation: (This is an example of a randomised question - the next time you use this example you will probably be given a different calculation to do):
\n$\\var{a}\\times\\var{b}+\\var{c}=\\;$[[0]]
\nYou have to input a whole number - it could be in decimal form. If the answer was $2$ then you could input 2 or 2.0 - try both forms.
Decimals
\nMany calculations will result in numbers which need to be entered in decimal notation, and the question will ask for a certain number of decimal places.
\nOften there is a small tolerance built in so that if you get the result wrong by 1 in the last decimal place then it will be marked as correct. But accuracy is important, so make sure that you get the calculations correct.
\nFor example:
\nInput $\\displaystyle \\frac{\\var{a1}}{\\var{b1}}$ as a decimal correct to 2 decimal places here: [[0]]
\nTry entering the correct value and submitting. Then vary the last decimal place by 1 either way and submitting, and then the last place by 2 either way and submitting.
\nTry putting in the fraction as it is (i.e. $\\var{a1}/\\var{b1}$ ) and see what happens.
\nThe system gives an error message as what you have put in is not a direct representation of a number. But you can always re-enter.
\nSo be careful - always check after submitting your answer that the input field contains the answer that you thought you entered.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{ans1-tol}", "maxValue": "{ans1+tol}", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Fractions
\nYou will find that some questions may ask you to input fractions and not decimals.
\nFor example, find the following sum as a fraction:
\n$\\displaystyle \\frac{1}{\\var{a1}}+\\frac{1}{\\var{b1}}=\\;$[[0]]
\n(input as a fraction and not a decimal)
\nHint: the answer is {a1+b1}/{a1*b1}
\nTry inputting the decimal version of this to as many places as you like (for example given by the calculator on the PC - you can copy this from the calculator and paste into the input field) and see what happens.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{a1+b1}/{a1*b1}", "vsetRangePoints": 5, "notallowed": {"message": "Simplify into a single fraction. Do not enter as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["+", "."]}, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": [], "variableReplacements": [], "failureRate": 1, "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "all, fractionNumbers", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "prompt": "As this question is in practice mode, if you click on the Reveal answers button all of the question fields are filled with the correct answers. Also, if available, there will be a full solution given under the heading Advice. Just scroll down to see this. However, there is no advice available for this question as it is not needed.
\nFinally as you are in practice mode, if you click on the Try another question like this one button at the bottom you will get this question again but with different numbers (usually!), and you can try it again. This is true for all practice mode questions which are randomised.
", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "unitTests": []}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example we show how to enter numbers, either as
\nDetails on inputting numbers into Numbas.
"}, "advice": "No advice available.
"}, {"name": "How to enter numbers and algebraic symbols - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {}, "ungrouped_variables": [], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Questions are often split into parts. In each part you will see various input fields for your answers.
\nThis is the first part and contains one question for you to answer. It will be clear from the question what you need to enter in each field.
\nFor example, a question could be:
\n$2+2=\\;$[[0]] (enter a number)
\nYou are expected to enter the answer and then press the Submit part button. Try it. Enter the correct value and press Submit part - a tick appears. Brilliant!!
\nNow enter an incorrect value. Press Submit part and a cross appears. Note the feedback underneath the button - in this case there is not much to say!
\nThis is the sort of feedback you get in practice mode.
\nTry putting in 2+2 as your answer and see what happens as well. You will be given an error message; click on OK and continue. So you must be careful and always check that the answer in the input field is what you expect it to be before you move on.
Pressing the Reveal answers button gives you the answers for all parts and usually also gives you a full solution for each part. This is only available in practice mode and certainly not available in a real assessment.
\nAlso note that in practice mode you have available a button at the bottom, Try another question like this one. This is useful for you to try other versions of the question. This question is not randomised, so you will get the same one back again!
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "4", "maxValue": "4", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "This is the second part of this example and it contains 2 questions. You enter your answers for both and then press Submit part.
\nNote that your input in mathematical notation is displayed next to your input so you can check it has been interpreted correctly.
\n$x+x=\\;$[[0]] (Enter a multiple of $x$ )
\nEnter your answer as 2*x. You could just enter 2x without the * for all multiplications as there are some cases where it is necessary in order to avoid ambiguity.
Simplify the following expression. Once again you see your input rendered in the best possible mathematical notation next to where you type. This check becomes more important when you input more complicated expressions.
\n$2x-x+y-2y=\\;$[[1]]
\nTry getting one right and one wrong and see the sort of feedback you get (the grey tick indicates that you have some, but not all, of the available marks). Also try inputting x+x for the answer to the first question in this part and see what happens after you submit.
Note the red exclamation marks next to the input field when you enter something the system does not like or you have pressed Submit part without answering the question. Move the cursor over the mark and you will get a message saying what the problem is.
\nThe Submit all parts button at the bottom allows you to answer everything in the question at once without submitting each part separately. In this case, the answers in both parts will be submitted.
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "2*x", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "notallowed": {"showStrings": false, "message": "Simplify the expression please!
", "strings": ["+"], "partialCredit": 0}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "marks": 1, "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}, {"answer": "x-y", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "notallowed": {"showStrings": false, "message": "Simplify further!
", "strings": ["+"], "partialCredit": 0}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "marks": 1, "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "This example explains how you enter your answers and submit them.
\nThis example and the others are in practice mode - you will be given information on whether or not you have the answer correct or not.
\nIf you use Numbas for a real assessment, it does not give you this information.
\nIt is very important that you submit all your answers. If you do not your results will not be recorded. Note that the list of questions in the exam on the left of the window gives information on whether or not you have completed a question.
\nGo to the next question. You can then come back. Note that until you quit the exam for good you can go back to any question and change your answers if you want to.
", "tags": ["answers", "checked2015", "entering", "fields", "input", "introduction", "mathematical notation", "Numbas", "numbas", "parts", "practice mode", "reveal", "submit"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Entering numbers and algebraic symbols in Numbas.
"}, "advice": ""}, {"name": "How to enter powers - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"scripts": {}, "gaps": [{"answer": "e^({a+b}*x)", "vsetrange": [0, 0.1], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "To test your input of powers try the following examples:
\nInput as a single power of $x$:
\n$\\simplify[all]{e^({a}*x)e^({b}*x)}=\\;$[[0]]
\n(The answer is $\\simplify[all]{e^({a+b}x)}$ but you have to enter it properly.)
\nYour input is shown in mathematical notation in a box next to your input so that you can check that you have entered it correctly.
\nClick on Submit part to check on your answer.
\nClick on the input field and edit your answer by inputting without brackets around the powers to see what happens.
\n\n
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^({c+d})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "
Input $x^{\\var{c}}x^{\\var{d}}$ as a single power of $x$.
\nFor example, you would input $x^{-6}x^{-5}$ as x^(-11).
$x^{\\var{c}}x^{\\var{d}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^{f}*y^{f}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input in the form $x^a*y^b$ for suitable values of $a$ and $b$.
", "showStrings": false, "partialCredit": 0, "strings": ["xy", "x*y"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Input $(x \\cdot y)^{\\var{f}}$ in the form $x^a \\times y^b$ for suitable values of $a$ and $b$.
\n$(x \\cdot y)^{\\var{f}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12 except a)*s", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12)*s", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..-6)", "description": "", "name": "c"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..-6)", "description": "", "name": "d"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5,-4,-3,-2,-1)", "description": "", "name": "f"}}, "ungrouped_variables": ["a", "c", "b", "d", "f", "s"], "variable_groups": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example we show you how to input powers. It is important that you get this right as many questions ask for such inputs.
\nThe standard way of inputting powers is as follows:
\n$a^b$ is input as a^b - and this is the only way to input powers.
But you have to be careful with inputting expressions such as $e^{2x}$ and $(xy)^2$. In these cases brackets should be used, as we now show:
\n| Power | Correct Input | Incorrect Input |
|---|---|---|
| $e^{2x}$ | \ne^(2*x) | \ne^2*x (system thinks this is $e^2 \\times x$) | \n
| $(xy)^2$ | \n(x*y)^2 | \nx*y^2 (system thinks this is $x \\times y^2$) | \n
So make sure that you use brackets to properly define your powers. This is a major source of input inaccuracies.
", "tags": ["brackets", "checked2015", "input", "introduction", "Numbas", "numbas", "powers"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Information on inputting powers
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\nSelect:
\nThis variable was created solely for the purpose of being able to publish this question.
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", "ii) $x^2$ is greater than $x$
", "iii) If $x$ is negative, $x^2$ is negative
", "iv) If $x$ is negative, $x^3$ is negative
", "v) $x^2 = x$
", "vi) $x^2 = - x$
", "vii) $(x+1)^2 \\gt x$
", "viii) $(x+1)^3 \\gt x$
", "ix) $x^3 \\times x = x^2 \\times x^2$
", "x) $x^2$ has the opposite sign to $x$
", "xi) $x^3$ has the opposite sign to $x$
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"}, "preamble": {"css": "", "js": ""}, "advice": "i)
\nSuppose $x$ is negative, for example $x = -5$.
\n\\[ \\begin{align} x^2 &= (-5)^2 \\\\&= 25 \\\\ x^3 &= (-5)^3 \\\\&= -125 \\\\x^3 &\\lt x^2\\text{.} \\end{align} \\]
\nNow let $x$ be positive, for example $x = 5$.
\n\\[ \\begin{align} x^2 &= 5^2 \\\\&= 25 \\\\ x^3 &= 5^3 \\\\&= 125 \\\\x^3 &\\gt x^2\\text{.}\\end{align} \\]
\nTherefore, $x^3$ is sometimes greater than $x^2$.
\n\n
\n
ii)
\nThis is true for either $x \\gt 1$ or $x \\lt 0$ but false for $0 \\leq x \\leq 1$. For example, let $x=0.5$. Then
\n\\[ \\begin{align} \\text{When } x &= 0.5\\text{,} \\\\x^2 &= 0.25\\text{, so} \\\\ x^2 &\\lt x \\text{.} \\end{align} \\]
\nTherefore, $x^2$ is sometimes greater than $x$.
\n\n
\n
iii)
\nMultiplying two negative numbers gives a positive answer and multiplying two postive numbers gives a positive answer. Therefore, $x^2$ is never negative.
\n\n
\n
iv)
\nMultiplying a negative number by itself an odd number of times always gives a negative answer. For example, let $x = -1$. Then
\n\\[\\begin{align} x^3 &= (-1)^3 \\\\&= -1\\times-1\\times-1 \\\\&=1\\times-1 \\\\&= - 1 \\text{.} \\end{align}\\]
\nTherefore, if $x$ is negative, $x^3$ is always negative.
\n\n
\n
v)
\nThis is true when $x = 1$ but false for all other values of $x$. Therefore, $x^2$ sometimes equals $x$.
\n\n
\n
vi)
\nThis is true when $x = -1$ but false for all other values of $x$. Therefore, $x^2$ sometimes equals $-x$.
\n\n
\n
vii)
\nWhen $x$ is positive, for example $x = 5$:
\n\\[ \\begin{align} (x+1)^2 &= (5 + 1)^2 \\\\&= 6^2 \\\\&= 36 \\gt x = 5 \\end{align} \\]
\nWhen $x = 0$:
\n\\[ \\begin{align} (x+1)^2 &= (0 + 1)^2 \\\\&= 1^2 \\\\&= 1 \\gt x = 0 \\end{align} \\]
\nWhen $x$ is negative, such as $x = -4$:
\n\\[ \\begin{align} (x+1)^2 &= (-4 + 1)^2 \\\\&= (-3)^2 \\\\&= 9 \\gt x = -4 \\end{align} \\]
\nTo see the behaviour of $(x+1)^2$ a bit more clearly, we make a table for values $-3 \\leq x \\leq 3$:
\n| $x$ | \n$-3$ | \n$-2$ | \n$-1$ | \n$-0.5$ | \n$0$ | \n$0.5$ | \n$1$ | \n$2$ | \n\n $3$ \n | \n
|---|---|---|---|---|---|---|---|---|---|
| $(x+1)^2$ | \n$4$ | \n$1$ | \n$0$ | \n$0.25$ | \n$1$ | \n$2.25$ | \n$4$ | \n$9$ | \n$16$ | \n
Therefore, $(x+1)^2$ is always greater than $x$.
\n\n\n
viii)
\nWhen $x$ is positive, for example $x = 5$:
\n\\[ \\begin{align} (x+1)^3 &= (5 + 1)^3 \\\\&= 6^3 \\\\&= 216 \\gt x = 5 \\end{align} \\]
\nWhen $x = 0$:
\n\\[ \\begin{align} (x+1)^3 &= (0 + 1)^3 \\\\&= 1^3 \\\\&= 1 \\gt x = 0 \\end{align} \\]
\nWhen $x$ is negative, such as $x = -4$:
\n\\[ \\begin{align} (x+1)^2 &= (-4 + 1)^3 \\\\&= (-3)^3 \\\\&= -27 \\lt x = -4 \\end{align} \\]
\nTo see the behaviour of $(x+1)^3$ a bit more clearly, we make a table for values $-3 \\leq x \\leq 3$:
\n| $x$ | \n$-3$ | \n$-2$ | \n$-1$ | \n$-0.5$ | \n$0$ | \n$0.5$ | \n$1$ | \n$2$ | \n\n $3$ \n | \n
|---|---|---|---|---|---|---|---|---|---|
| $(x+1)^3$ | \n$-8$ | \n$-1$ | \n$0$ | \n$0.125$ | \n$1$ | \n$3.375$ | \n$8$ | \n$27$ | \n$64$ | \n
Therefore, $(x+1)^3$ is sometimes greater than $x$.
\n\n\nix)
\nWe can write
\n\\[x^3\\times x = x \\times x \\times x \\times x = x^2 \\times x^2\\text{.}\\]
\nTherefore, $x^3 \\times x$ always equals $x^2 \\times x^2$.
\n\n\nx)
\nSince $x^2$ is always positive, $x^2$ only has the opposite sign to $x$ when $x$ is negative.
\nTherefore, $x^2$ sometimes has the opposite sign to $x$.
\n\n\nxi)
\nAs seen in part iv), multiplying a negative number by itself an odd number of times always gives a negative answer. It is also true that multiplying a positive number by itself an odd number of times will give a positive answer. So, $x^3$ always has the same sign as $x$, since we have an odd power.
\nTherefore, $x^3$ never has the opposite sign to $x$.
\n\n"}, {"name": "Arithmetic operations: Inequalities", "extensions": [], "custom_part_types": [], "resources": [["question-resources/drawingresize_grbP9s8.svg", "/srv/numbas/media/question-resources/drawingresize_grbP9s8.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "type": "question", "statement": "
Choose the correct symbols to describe the relations between each of these pairs of numbers.
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$\\var{dec[6] + 0.001 + random[0]}$ [[0]] $\\var{dec[6] - random[1]}$
\n", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["<
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$\\var{neg[7] + random2}$ [[0]] $\\var{neg[7] + 0.9 + random[2]}$
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$(\\var{neg[3]}) \\times (\\var{neg[2]})$ [[0]] $\\var{-neg[3]*neg[2]}$
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"}, "preamble": {"css": "", "js": ""}, "advice": "\\[\\begin{align} \\text{Symbol }&\\lt \\text{ denotes \"less than\".} \\\\ \\text{Symbol }&\\gt \\text{denotes \"greater than\".} \\end{align}\\]
\n$\\var{dec[6] + 0.001 + random[0]}$ is greater than $\\var{dec[6] - random[1]}$ so
\n\\[\\var{dec[6] + 0.001 + random[0]} \\gt \\var{dec[6] - random[1]} \\text{.} \\]
\nWhen both of the numbers that you are comparing are negative, it may be tempting to ignore the negative signs and make an incorrect assumption. For example, when we have -5 and -4 we might ignore the signs and assume -5 is larger than -4 since +5 is larger than +4. This is however wrong, -5 < -4.
\nTo understand this a bit better, look at the following number line:
\n\nFollowing the number line from left to right, we can see that $\\var{neg[7] + random2}$ is less than $\\var{neg[7] + 0.9 + random[2]}$, so
\n\\[\\var{neg[7] + random2} \\lt \\var{neg[7] + 0.9 + random[2]} \\text{.}\\]
\n\nMultiplying two negative numbers results in a positive number. Therefore we can see without performing any calculation that $(\\var{neg[3]}) \\times (\\var{neg[2]}) \\gt \\var{-neg[3]*neg[2]}$ as positive numbers are always larger than negative numbers.
\n\\[(\\var{neg[3]} \\times \\var{neg[2]}) \\gt \\var{-neg[3]*neg[2]}\\]
\n\\[\\var{neg[3] * neg[2]} \\gt \\var{-neg[3]*neg[2]}\\]
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", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "advice": "When given a number in the form $A \\times 10^n$, we can think of $n$ as a number telling us how many places to move the decimal point.
\nWhen $n$ is positive, we move the decimal point to the right side, for example:
\n\\[ 1.5 \\times 10^3 = 1500.0 \\text{ .} \\]
\nWhen $n$ is negative, we move the decimal point to the left side, for example:
\n\\[ 1.5 \\times 10^{-3} = 0.0015 \\text{ .} \\]
\nWhen $n = 0$, we do not move the decimal point:
\n\\[ 1.5 \\times 10^0 = 1.5 \\text{ .}\\]
\n\n
In $\\var{A[0]} \\times 10^\\var{ran}$, $n = \\var{ran}$ and so we move the decimal point {ran} places to the right.
\n\\[\\var{A[0]} ⇒ \\var{precround((A[0] * 10^ran), 0)}\\]
\n\nIn $\\var{A[1]} \\times 10^\\var{-ran + 4}$, $n = \\var{-ran +4}$ and so we move the decimal point {ran -4} places to the left.
\n\\[\\var{A[1]} ⇒ \\var{A[1]*10^(-ran+4)}\\]
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\n", "customName": "", "gaps": [{"answer": "{A[1]*10^(-ran+4)}", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "valuegenerators": [], "variableReplacements": [], "unitTests": [], "customName": "", "checkingType": "absdiff", "vsetRange": [0, 1], "checkVariableNames": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "type": "jme", "mustmatchpattern": {"message": "", "nameToCompare": "", "partialCredit": 0, "pattern": "$n"}, "showPreview": true, "vsetRangePoints": 5, "useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": "0.5", "scripts": {}}], "useCustomName": false, "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "tags": ["conversion", "converting", "standard form", "standard index form", "taxonomy"], "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Decimals to fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}], "variable_groups": [{"variables": ["a", "b", "c", "d", "b_", "b_coprime", "d_coprime", "answer", "cround"], "name": "Part a"}, {"variables": ["f", "f2", "mygcd", "h", "j", "k", "f3", "f4", "f1000", "numerator", "g", "gcd1", "numerator_coprime", "g_coprime"], "name": "Part b"}], "functions": {}, "rulesets": {}, "ungrouped_variables": [], "metadata": {"description": "Identify well-known fractional equivalents of decimals. Convert obscure decimals and recurring decimals into fractions.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "To convert a decimal into a fraction, firstly place the decimal over $1$ as a fraction, and then multiply both the numerator and denominator by $10$ for however many decimal places the decimal has. For example, if the decimal was $0.1$, you would multiply the fraction by $10$ as there is one decimal place. If the decimal was $0.01$, you would multiply it by $100$, as there are two decimal places.
\ni)
\n$\\var{a}$
\n\\[
\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}
\\]
ii)
\n$\\var{b}$
\n\\[
\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}
\\]
iii)
\n\n$\\var{d}$
\n\\[
\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}
\\]
iv)
\n\n$0.\\dot{\\var{c}}$
\nTo convert a recurring decimal to a fraction, the first step is to set up a simple equation where
\n\\[
x=0.\\dot{\\var{c}}\\text{.}
\\]
By multiplying both sides by $10$, we can gain another simple equation where
\n\\[
10x=\\var{c}.\\dot{\\var{c}}\\text{.}
\\]
By subtracting one equation from the other, we can find the fraction equivalent of the recurring decimal.
\n\\[
\\begin{align}
&&\\var{c}.\\dot{\\var{c}}&={10}x\\\\
-&&{0.\\dot{\\var{c}}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}\\\\
&&{\\var{c}}&=9x\\\\
\\\\
&&\\frac{\\var{c}}{9}&=x
\\end{align}
\\]
$\\displaystyle\\frac{\\var{c}}{9}$ simplifies to $\\simplify{{{c}}/{9}}$ by dividing by $3$ and therefore, $0.\\dot{\\var{c}}=\\simplify{{{c}}/{9}}$ in its fractional form.}
$\\displaystyle\\var{f}$
\n\\[
\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}
\\]
From this, we can look to see if we can cancel down the fraction by finding the highest common divisor between the numerator and denominator. This is $\\var{mygcd}$.
\nTherefore, it is not possible to simplify the answer any further and the final answer is
\nSimplifying by this amount gives the final answer
\n\\[\\frac{\\var{f3}}{\\var{f4}}.\\]
\n$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$
\nTo convert a recurring decimal to a fraction, the first step is to set up a simple equation where,
\n$x=\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$
\n
By multiplying both sides by $100$ to isolate the recurring section on the left hand side of the decimal point, we can gain another simple equation
$100x=\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}.$
\n\nNow that we have two equations in terms of $x$, we can subtract one from the other and solve to get a value of $x$.
\n\\[
\\begin{align}
&&\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}&=100x\\\\
-&&\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}
\\\\
&&{{\\var{h}}\\var{j}\\var{k-h}}&=99x\\\\
\\\\
&&\\frac{\\var{numerator}}{\\var{g}}&=x\\text{.}\\\\
\\end{align}
\\]
From this, we should look to see if it is possible to simplify by finding the greatest common divisor of the numerator and the denominator. The greatest common divisor is $\\var{gcd1 }.$
\nTherefore, it is not possible to simplify and so
\nSimplifying by this value gives the fraction $\\displaystyle\\simplify{{{numerator}}/{g}}$ and so
\n\\[
\\begin{align}
\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ in its fractional form.}\\\\
\\end{align}
\\]
Fractions can be equivalently represented as decimals and vice versa. One form may be more useful in a context than another and it is useful to practise how to change between them.
\nHave a go at these questions involving fractions and decimals, remembering to write your answer in its simplest form.
", "preamble": {"css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n", "js": ""}, "variables": {"answer": {"templateType": "anything", "description": "", "name": "answer", "group": "Part a", "definition": "round(10c-c)"}, "cround": {"templateType": "anything", "description": "", "name": "cround", "group": "Part a", "definition": "c/3"}, "g": {"templateType": "anything", "description": "", "name": "g", "group": "Part b", "definition": "99"}, "j": {"templateType": "anything", "description": "", "name": "j", "group": "Part b", "definition": "random(1..9)"}, "g_coprime": {"templateType": "anything", "description": "", "name": "g_coprime", "group": "Part b", "definition": "g/gcd1"}, "d": {"templateType": "anything", "description": "", "name": "d", "group": "Part a", "definition": "random(0.2,0.4,0.6,0.8)"}, "a": {"templateType": "anything", "description": "", "name": "a", "group": "Part a", "definition": "random(0.1,0.3,0.7,0.9)"}, "gcd1": {"templateType": "anything", "description": "", "name": "gcd1", "group": "Part b", "definition": "gcd(numerator,g)"}, "h": {"templateType": "anything", "description": "", "name": "h", "group": "Part b", "definition": "random(1..5)"}, "b": {"templateType": "anything", "description": "", "name": "b", "group": "Part a", "definition": "random(0.25,0.75)"}, "f4": {"templateType": "anything", "description": "", "name": "f4", "group": "Part b", "definition": "f1000/gcd(f2,f1000)"}, "d_coprime": {"templateType": "anything", "description": "", "name": "d_coprime", "group": "Part a", "definition": "10d/gcd(10d,10)"}, "f1000": {"templateType": "anything", "description": "", "name": "f1000", "group": "Part b", "definition": "1000"}, "numerator_coprime": {"templateType": "anything", "description": "", "name": "numerator_coprime", "group": "Part b", "definition": "numerator/gcd1"}, "c": {"templateType": "anything", "description": "", "name": "c", "group": "Part a", "definition": "random(3,6)"}, "k": {"templateType": "anything", "description": "", "name": "k", "group": "Part b", "definition": "random(h..9 except j except h)"}, "f2": {"templateType": "anything", "description": "", "name": "f2", "group": "Part b", "definition": "precround(f1000*f,0)"}, "f3": {"templateType": "anything", "description": "", "name": "f3", "group": "Part b", "definition": "f2/gcd(f2,f1000)"}, "numerator": {"templateType": "anything", "description": "", "name": "numerator", "group": "Part b", "definition": "h*100+j*10+k - h"}, "mygcd": {"templateType": "anything", "description": "", "name": "mygcd", "group": "Part b", "definition": "gcd(f2,f1000)"}, "b_": {"templateType": "anything", "description": "", "name": "b_", "group": "Part a", "definition": "gcd(100*b,100)"}, "f": {"templateType": "anything", "description": "", "name": "f", "group": "Part b", "definition": "random(0.1..0.8#0.002)"}, "b_coprime": {"templateType": "anything", "description": "", "name": "b_coprime", "group": "Part a", "definition": "100*b/b_"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "Express these common decimals as their fraction equivalent.
\ni)
\n$\\var{a}=$
ii)
\n$\\var{b}=$
iii)
\n$\\var{d}=$
iv)
\n$0.\\dot{\\var{c}}=$
Convert this decimal to a fraction, giving your answer in its simplest form.
\n$\\displaystyle\\var{f} = $
", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": "2", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "f3", "allowFractions": false, "minValue": "f3", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "2", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "f4", "allowFractions": false, "minValue": "f4", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "prompt": "
Convert these decimals to a fraction, giving your answer in its simplest form.
\nii)
\n$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}} = $
Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions.
", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": [], "advice": "When faced with dividing fractions, it much easier to switch one of the fractions around and multiply them together instead of divide them.
\n\\[ \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}} \\right) \\equiv \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\times\\frac{\\var{j_coprime}}{\\var{h_coprime}} \\right) = \\frac{\\var{fj}}{\\var{gh}} \\]
\nThen, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd1}$.
\nThis gives a final answer of $\\displaystyle\\simplify{{fj}/{gh}}$.
\n\n\\[ \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}} \\equiv \\left( \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\times\\frac{\\var{j1_coprime}}{\\var{h1_coprime}} \\right)=\\frac{\\var{f1j1}}{\\var{g1h1}} \\]
\nThen, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd2}$.
\nThis gives a final answer of $\\displaystyle\\simplify{{f1j1}/{g1h1}}$.
\n\n\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}} \\]
\nThe first thing to do is to change the mixed numbers into improper fractions.
\nAn improper fraction is a fraction where the numerator is greater than the denominator. To change a mixed fraction to an improper fraction, multiply the integer part of the mixed number by the denominator, and add it to the existing numerator to make the new numerator of the improper fraction. The denominator will stay the same.
\n\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\equiv\\frac{(\\var{f3}\\times\\var{h3_coprime})+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3}+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}} \\]
\n\\[ {\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}\\equiv\\frac{(\\var{f4}\\times\\var{h4_coprime})+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4}+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]
\nWe now have our mixed numbers as improper fractions.
\n\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]
\nNow, use the same method as in parts a) and b) to divide by switching around one fraction and changing the division symbol to multiplication.
\n\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}}\\equiv\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\times\\frac{\\var{h4_coprime}}{\\var{f4h4+g4_coprime}}=\\frac{\\var{num}}{\\var{denom}} \\]
\nFinally, the last thing to do is to simplify your answer down by finding the highest common divisor in the numerator and denominator, which in this case is $\\var{gcd3}$.
\nBy doing this, you will get a final answer of
\n\\[ \\simplify{{num}/{denom}} \\]
\n\\[ \\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} \\]
\nConsider the denominator first, as following the rules of BODMAS, you should address brackets first.
\nYou need to get a common denominator for both terms on the denominator, like this:
\n\\[ \\var{b}\\times\\frac{\\var{d}}{\\var{d}} = \\frac{\\var{bd}}{\\var{d}} \\]
\nThis now allows you to complete the addition or subtraction as both terms have a common denominator.
\n\\[ {\\simplify[all,!collectNumbers]{{bd}/{d}-{c}/{d}}} = \\frac{\\var{bd_c}}{\\var{d}} \\]
\nThis means that the expression is now:
\n\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}} \\]
\nDealing with this requires a bit of manipulation. You have to change the divisor of the denominator to be a mulitplier of the numerator. The denominator, ${\\var{bd_c}}$ was being divided by ${\\var{d}}$ but by flipping it around, the numerator, ${\\var{a}}$ will be mulitplied by ${\\var{d}}$. The value of the expression remains the same.
\n\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}}\\equiv \\frac{(\\var{a})\\times(\\var{d})}{\\var{bd_c}}= \\frac{\\var{ad}}{\\var{bd_c}} \\]
\nFrom this, you can try to cancel the expression down by finding the highest common factor of the numerator and denominator, to give a final answer of
\n\\[ \\simplify{{ad}/{bd_c}} \\]
", "variable_groups": [{"name": "part d", "variables": ["a", "b", "c", "d", "bd", "ad", "gcd", "ad_gcd", "bcd_gcd", "bd_c"]}, {"name": "part a", "variables": ["f", "g", "f_coprime", "g_coprime", "h", "j", "h_coprime", "j_coprime", "fj", "gh", "gcd1"]}, {"name": "part b", "variables": ["f1", "g1", "f1_coprime", "g1_coprime", "h1", "j1", "h1_coprime", "j1_coprime", "f1j1", "g1h1", "gcd2"]}, {"name": "part c", "variables": ["f3", "g3", "h3", "g3_coprime", "h3_coprime", "f4", "g4", "h4", "g4_coprime", "h4_coprime", "f3h3", "f4h4", "num", "denom", "gcd3"]}], "statement": "Evaluate the following sums involving division of fractions. Simplify your answers where possible.
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}}=$
$\\displaystyle\\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}}=$
$\\displaystyle{\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}=$
$\\displaystyle\\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} =$
variable f4 times h4.
\nUsed in part c)
", "group": "part c", "definition": "f4*h4_coprime", "name": "f4h4", "templateType": "anything"}, "g4_coprime": {"description": "PART C
", "group": "part c", "definition": "g4/gcd(g4,h4)", "name": "g4_coprime", "templateType": "anything"}, "h4": {"description": "Random number but not the same number as variable g4.
\nUsed in part c.
", "group": "part c", "definition": "random(5..8 except g4)", "name": "h4", "templateType": "anything"}, "g": {"description": "Random number between 2 and 10 and not the same number as variable f.
\nUsed in part a).
", "group": "part a", "definition": "random(f..12 except f) ", "name": "g", "templateType": "anything"}, "a": {"description": "Random number between 1 and 20
\nUsed by part d)
", "group": "part d", "definition": "random(1..10#1)", "name": "a", "templateType": "randrange"}, "bd_c": {"description": "Unsimplified denominator for part d).
", "group": "part d", "definition": "(bd-c)", "name": "bd_c", "templateType": "anything"}, "h3_coprime": {"description": "PART C
", "group": "part c", "definition": "h3/gcd(g3,h3)", "name": "h3_coprime", "templateType": "anything"}, "f_coprime": {"description": "PART A
", "group": "part a", "definition": "f/gcd(f,g)", "name": "f_coprime", "templateType": "anything"}, "g_coprime": {"description": "PART A
", "group": "part a", "definition": "g/gcd(f,g)", "name": "g_coprime", "templateType": "anything"}, "j1_coprime": {"description": "PART B
", "group": "part b", "definition": "j1/gcd(h1,j1)", "name": "j1_coprime", "templateType": "anything"}, "gcd2": {"description": "greatest common divisor of variables f1j1 and g1h1.
\nUsed in part b).
", "group": "part b", "definition": "gcd(f1j1,g1h1)", "name": "gcd2", "templateType": "anything"}, "c": {"description": "Random prime number between -10 and 10.
\nUsed by part d).
", "group": "part d", "definition": "random([-7,-5,-3,-2,-1,1,2,3,5,7] except d)", "name": "c", "templateType": "anything"}, "ad_gcd": {"description": "Correct answer for the numerator in part d)
", "group": "part d", "definition": "ad/gcd", "name": "ad_gcd", "templateType": "anything"}, "g1_coprime": {"description": "PART B
", "group": "part b", "definition": "g1/gcd(f1,g1)", "name": "g1_coprime", "templateType": "anything"}, "h1_coprime": {"description": "PART B
", "group": "part b", "definition": "h1/gcd(h1,j1)", "name": "h1_coprime", "templateType": "anything"}, "gcd3": {"description": "greatest common denominator for part c.
", "group": "part c", "definition": "gcd(num,denom)", "name": "gcd3", "templateType": "anything"}, "bd": {"description": "Variable b times variable d.
\nUsed in part d)
", "group": "part d", "definition": "b*d", "name": "bd", "templateType": "anything"}, "j1": {"description": "Random number between 2 and 20 and not the same value as variable h1.
\nUsed in part b).
", "group": "part b", "definition": "random(h1..11 except h1)", "name": "j1", "templateType": "anything"}, "g1h1": {"description": "variable g1 times h1.
\nUsed in part b).
", "group": "part b", "definition": "g1_coprime*h1_coprime", "name": "g1h1", "templateType": "anything"}, "f": {"description": "Random number between 2 and 10.
\nUsed in part a).
", "group": "part a", "definition": "random(2..10)", "name": "f", "templateType": "anything"}, "b": {"description": "Random number between 1 and 10.
\nUsed by part d)
", "group": "part d", "definition": "random(1..10#1)", "name": "b", "templateType": "randrange"}, "bcd_gcd": {"description": "Correct answer for the denominator in part d).
", "group": "part d", "definition": "{bd_c}/gcd", "name": "bcd_gcd", "templateType": "anything"}, "f4": {"description": "Random number.
\nUsed in part c).
", "group": "part c", "definition": "random(1..3)", "name": "f4", "templateType": "anything"}, "f1": {"description": "Random number between 2 and 20.
\nUsed in part b)
", "group": "part b", "definition": "random(2..10)", "name": "f1", "templateType": "anything"}, "d": {"description": "Random prime number between 10 and 20.
\nUsed in part d).
", "group": "part d", "definition": "random(7,11,13,17)", "name": "d", "templateType": "anything"}, "g3": {"description": "Random number.
\nUsed in part c).
", "group": "part c", "definition": "random(1..3)", "name": "g3", "templateType": "anything"}, "f3h3": {"description": "variable f3 times h3.
", "group": "part c", "definition": "f3*h3_coprime", "name": "f3h3", "templateType": "anything"}, "h": {"description": "Random number from 2 to 10.
\nUsed in part a).
", "group": "part a", "definition": "random(2..10)", "name": "h", "templateType": "anything"}, "gh": {"description": "variable g times variable h.
\nUsed in part a).
", "group": "part a", "definition": "g_coprime*h_coprime", "name": "gh", "templateType": "anything"}, "j_coprime": {"description": "PART A
", "group": "part a", "definition": "j/gcd(h,j)", "name": "j_coprime", "templateType": "anything"}, "denom": {"description": "Unsimplified denominator of part c.
", "group": "part c", "definition": "h3_coprime*(f4h4+g4_coprime)", "name": "denom", "templateType": "anything"}, "j": {"description": "Random number between 2 and 10 and not the same value as h.
\nUsed in part a).
", "group": "part a", "definition": "random(h..12 except h)", "name": "j", "templateType": "anything"}, "f1j1": {"description": "variable f1 times j1.
\nUsed in part b).
", "group": "part b", "definition": "f1_coprime*j1_coprime", "name": "f1j1", "templateType": "anything"}, "h4_coprime": {"description": "PART C
", "group": "part c", "definition": "h4/gcd(g4,h4)", "name": "h4_coprime", "templateType": "anything"}, "g1": {"description": "Random number between 2 and 30 and not the same value as variable f1.
\nUsed in part b).
", "group": "part b", "definition": "random(f1..11 except f1) ", "name": "g1", "templateType": "anything"}, "fj": {"description": "variable f times variable j.
\nUsed in part a).
", "group": "part a", "definition": "f_coprime*j_coprime", "name": "fj", "templateType": "anything"}, "gcd": {"description": "Greatest common divisor of ad and bd_c.
\nUsed in part d).
", "group": "part d", "definition": "gcd(ad,bd_c)", "name": "gcd", "templateType": "anything"}, "f3": {"description": "Random number between 2 and 6.
\nUsed in part c).
", "group": "part c", "definition": "random(1..3#1)", "name": "f3", "templateType": "randrange"}, "f1_coprime": {"description": "PART B
", "group": "part b", "definition": "f1/gcd(f1,g1)", "name": "f1_coprime", "templateType": "anything"}, "h3": {"description": "Random number and not the same value as variable g3.
\nUsed in part c).
", "group": "part c", "definition": "random(5..8)", "name": "h3", "templateType": "anything"}, "gcd1": {"description": "greatest common divisor of variable fj and gh.
\nUsed in part a).
", "group": "part a", "definition": "gcd(fj,gh)", "name": "gcd1", "templateType": "anything"}, "g3_coprime": {"description": "PART C
", "group": "part c", "definition": "g3/gcd(g3,h3)", "name": "g3_coprime", "templateType": "anything"}, "h_coprime": {"description": "PART A
", "group": "part a", "definition": "h/gcd(h,j)", "name": "h_coprime", "templateType": "anything"}, "g4": {"description": "Random number.
\nUsed in part c).
", "group": "part c", "definition": "random(1..5)", "name": "g4", "templateType": "anything"}, "h1": {"description": "Random number between 2 and 20.
\nUsed in part b).
", "group": "part b", "definition": "random(2..10)", "name": "h1", "templateType": "anything"}, "num": {"description": "numerator of the improper fraction in part c. Unsimplified.
", "group": "part c", "definition": "h4_coprime*(f3h3+g3_coprime)", "name": "num", "templateType": "anything"}, "ad": {"description": "Variable a times variable d.
\nUsed in part d).
", "group": "part d", "definition": "a*d", "name": "ad", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Using Surds, Rationalising the Denominator", "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/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "tags": ["Fractions", "fractions", "rationalise the denominator", "surds", "Surds", "taxonomy"], "metadata": {"description": "Manipulate surds and rationalise the denominator of a fraction when it is a surd.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "To include a square root sign in your answer use sqrt(). For example, to write $\\sqrt{3}$, type sqrt(3) into the answer box. If you are entering a number multiplied by the square root of some other number, for example $3\\sqrt{5}$, type 3*sqrt(5) into the answer box.
Surds can be manipulated using the rule
\n\\[\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}.\\]
\nWe are asked to state which of $\\sqrt{\\var{p}}$, $\\sqrt{\\simplify{{a}*{n}^2}}$, and $\\sqrt{\\var{a}}$ can be simplified further. Commonly, surds can be simplified if the number inside of the square root has a square number as a factor.
\nHere, $\\var{p}$ is a prime number which means that its only divisors are $\\var{p}$ and $1$.
\nTherefore, $\\sqrt{\\var{p}}$ cannot be simplified any further.
\nSimilarly, $\\var{a}$ is also a prime number, so $\\sqrt{\\var{a}}$ also cannot be simplified any further.
\nOn the other hand, $\\simplify{{a}*{n}^2}$ is not a prime number and we can use the previous rule to simplify $\\sqrt{\\simplify{{a}*{n}^2}}$ as
\n\\[
\\begin{align}
\\sqrt{\\simplify{{a}*{n}^2}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{a}}\\\\
&= \\simplify{{n}*sqrt({a})}.
\\end{align}
\\]
Using the same rule of manipulation as in part a), we can simplify $\\sqrt{\\simplify{{n}^2*{p}}}$ as
\n\\[
\\begin{align}
\\sqrt{\\simplify{{n}^2*{p}}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{p}}\\\\
&= \\simplify{{n}*sqrt({p})}.
\\end{align}
\\]
Here, we can use both of the rules for manipulating surds:
\n\\[\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b} \\text{.} \\]
\n\\[ \\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}} \\text{.} \\]
\nWe can simplify $\\displaystyle\\frac{ \\sqrt{\\simplify{{a}*{v}}} }{ \\sqrt{\\var{a}} }$ as follows.
\n\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\frac{\\sqrt{\\var{a}} \\times \\sqrt{\\var{v}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\times \\sqrt{\\var{v}} \\\\[0.5em]
&= \\simplify{{sqrt(a)/sqrt(a)}} \\times \\sqrt{\\var{v}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]
Or,
\n\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\sqrt{\\frac{\\simplify{{a}*{v}}}{\\var{a}}} \\\\[0.5em]
&= \\sqrt{\\var{v}} \\text{.}
\\end{align}
\\]
We can simplify the fraction as
\n\\[
\\begin{align}
\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}} &= \\frac{\\sqrt{\\simplify{({b*m})^2}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\frac{\\simplify{{b*m}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em]
&= \\simplify{{b}*sqrt({s})} \\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2{a})+{n}sqrt({b}^2*{a})} &= \\var{d}\\sqrt{\\var{a}} - \\var{b}(\\sqrt{\\simplify{{v}^2}} \\times \\sqrt{\\var{a}})+\\var{n}(\\sqrt{\\simplify{{b}^2}} \\times \\sqrt{\\var{a}}) \\\\
&= \\var{d}\\sqrt{\\var{a}} -\\var{b}(\\simplify{{v}*sqrt({a})})+\\var{n}(\\simplify{{b}*sqrt({a})}) \\\\
&= \\simplify{{d}sqrt({a})}-\\simplify{{b}*{v}sqrt({a})}+\\simplify{{n}*{b}sqrt({a})} \\\\
&= \\simplify{({d}-{b}*{v}+{n}*{b})sqrt({a})} \\text{.}
\\end{align}
\\]
We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{\\sqrt{a}}$, by multiplying the top and bottom by $\\sqrt{a}$.
\nTherefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$, we multiply top and bottom by $\\sqrt{\\var{a}}$.
\n\\[
\\begin{align}
\\frac{1}{\\sqrt{\\var{a}}} &= \\frac{1}{\\sqrt{\\var{a}}} \\times \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\\\[0.5em]
&= \\frac{\\sqrt{\\var{a}}}{\\var{a}} \\text{.}
\\end{align}
\\]
We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{a+\\sqrt{b}}$ by multiplying the top and bottom by $a-\\sqrt{b}$.
\nTherefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$, we multiply the top and bottom by $\\var{n} - \\sqrt{\\var{a}}$.
\n\\[
\\begin{align}
\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} &= \\frac{1}{\\var{n}+\\sqrt{\\var{a}}} \\times \\frac{\\var{n}-\\sqrt{\\var{a}}}{\\var{n}-\\sqrt{\\var{a}}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{(\\var{n}+\\sqrt{\\var{a}})(\\var{n}-\\sqrt{\\var{a}})} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2}-\\var{a}} \\\\[0.5em]
&=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2-{a}}} \\text{.}
\\end{align}
\\]
We rationalise the denominator of fractions of the form $\\displaystyle\\frac{1}{a-\\sqrt{b}}$ by multiplying the top and bottom by $a+\\sqrt{b}$.
\nTherefore, to rationalise the denominator of the fraction $\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}}$, we multiply the top and bottom by $\\var{d+p}+\\sqrt{\\var{p}}$.
\n\\[
\\begin{align}
\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} &= \\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} \\times \\frac{\\var{d+p}+\\sqrt{\\var{p}}}{\\var{d+p}+\\sqrt{\\var{p}}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{(\\var{d+p}-\\sqrt{\\var{p}})(\\var{d+p}+\\sqrt{\\var{p}})} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2}-\\var{p}} \\\\[0.5em]
&=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2-{p}}} \\\\[0.5em]
&=\\simplify{{t}/{(d+p)^2-p}}(\\var{d+p}+\\sqrt{\\var{p}}) \\\\[0.5em]
&= \\simplify[all,!noleadingMinus]{({t*(d+p)}+{t}*sqrt({p}))/({(d+p)^2-p})} \\text{.}
\\end{align}
\\]
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Input your answer here: [[0]]
No decimal numbers allowed.
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Input your answer as a fraction and not\n \n as a decimal.
Putting something here so Loughborough doesn't break.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Add/subtract fractions and reduce to lowest form.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
In this case we have:
\\[\\simplify[std,!unitFactor]{{a} / {g} + ({s1*b} / {f}) = ({a} * {f} + {g} * {s1*b}) / ({g} * {f}) ={a*f+s1*g*b}/{g*f}}.\\]
Note that this fraction is in its lowest form as there are no common factors in the denominator and the numerator.
Factorise $\\displaystyle{ax ^ 2 + bx + c}$ into linear factors.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Factorise the following quadratic expression $q(x)$ into linear factors i.e. input $q(x)$ in the form $(ax+b)(cx+d)$or of the form $a(x+b)(cx+d)$ for suitable integers $a$, $b$, $c$ and $d$ .
", "advice": "Direct Factorisation.
\nIf you can spot a direct factorisation then this is the quickest way to do this question.
\nFor this example we have the factorisation
\n\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]
\nFactorisation by finding the roots.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$.
\nThere are several methods of finding the roots – here are the main methods.
\nFinding the roots of a quadratic using the standard formula.
\nWe can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their common value is $-\\frac{b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\nFor this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify{{-(b*c+a*d)}^2-4*{a*b}*{c*d}={disc}}$
\n{rdis}.
\nSo the {rep} roots are:
\n\\[\\begin{eqnarray} x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 + n4}/ {n3}}\\\\ x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{(\\var{n1} - \\var{n4}) }{\\var{n3}} &=& \\simplify{{n1 - n4}/ {n3}} \\end{eqnarray}\\]
So we see that:
\\[q(x)=\\simplify{{a*b}}\\left(\\simplify{x-{n1 + n4}/ {n3}}\\right)\\left(\\simplify{x-{n1 - n4}/ {n3}}\\right)=\\simplify{({b} * x + { -d}) * ({a} * x + { -c})}\\]
Completing the square.
\nFirst we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1-abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
Finding these roots then gives the factorisation as before.
\\[q(x)=\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}\\]
$q(x)=\\;$ [[0]]
You can get more information on factorising a quadratic by clicking on Show steps. You will lose 1 mark if you do so.
", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "
Factorisation by finding the roots
\nIf you cannot spot a direct factorisation of a quadratic $q(x)$ then finding the roots of the equation $q(x)=0$ can help you.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ for some constant $a$.
\nFinding the roots of a quadratic using the standard formula
We can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
The two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their value is $\\frac{-b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "((({a} * x) + {( - c)}) * (({b} * x) + {( - d)}))", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.0001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "factorise the expression into two factors
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\n\\[\\log_a(x^{\\var{a1}}y^{\\var{b1}})=\\alpha\\log_a(x)+\\beta\\log_a(y)\\]
\n$\\alpha=\\;\\;$[[0]], $\\beta=\\;\\;$[[1]]
\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "((x ^ {f}) * (({c} * x) + {d}))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[\\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})=\\log_a(q(x))\\]
\n$q(x)=\\;\\;$[[0]]
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Answer the following questions on logarithms.
", "tags": ["checked2015", "log laws", "logarithm laws", "logarithmic expressions", "logarithms", "logs", "MAS1601", "mas1601", "rules for logarithms", "simplifying logarithms"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "2/06/2012:
\nAdded tags.
\nChanged statement to make question clearer.
\n19/07/2012:
\nAdded description.
\n25/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n17/08/2012:
\nMade copy to include in Simplify Algebraic Expressions exam.
", "licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\tExpress $\\log_a(x^{c}y^{d})$ in terms of $\\log_a(x)$ and $\\log_a(y)$. Find $q(x)$ such that $\\frac{f}{g}\\log_a(x)+\\log_a(rx+s)-\\log_a(x^{1/t})=\\log_a(q(x))$
\n \t\t\n \t\t"}, "advice": "
The rules for combining logs are
\n\\[\\begin{eqnarray*} \\log_a(bc)&=&\\log_a(b)+\\log_a(c)\\\\ \\\\ \\log_a\\left(\\frac{b}{c}\\right)&=&\\log_a(b)-\\log_a(c)\\\\ \\\\ \\log_a(b^r)&=&r\\log_a(b) \\end{eqnarray*} \\]
\na)
Using these rules gives:
\\[ \\begin{eqnarray*} \\log_a(x^{\\var{a1}}y^{\\var{b1}})&=&\\log_a(x^{\\var{a1}})+\\log_a(y^{\\var{b1}})\\\\ &=&\\var{a1}\\log_a(x)+\\var{b1}\\log_a(y) \\end{eqnarray*} \\]
b)
\\[\\begin{eqnarray*} \\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})&=&\\log_a(x^\\frac{\\var{a2}}{\\var{b2}})+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})\\\\ \\\\ &=&\\log_a\\left(\\simplify[std]{(x^({a2}/{b2})*({c}x+{d}))/(x^(1/{b2}))}\\right)\\\\ &=&\\log_a\\left(\\simplify{x^{f}*({c}x+{d})}\\right) \\end{eqnarray*} \\]
You must write your answer in the form p/q for integers p and q
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", "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"], "partialCredit": 0}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[\\simplify{{g} / ({a} + {s1} * ({b * g} / {f}))}\\]
Input your answer here: [[0]]
Your answer must be of the form a/b for suitable integers a and b. No decimal numbers allowed.
\nDo not include brackets in your answer.
", "marks": 0}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*f+s*b*g=1,-s,s)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(f=2,1,f=3,random(1,2),f=4,random(1,3),f=5, random(1..4),f=6,random(1,5),f=7,random(1..5),f=8,random(1,3,5),f=9,random(1,2,4,5),f=10,random(1,3),f=11,random(1..5))", "name": "b", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(g=2,random(3..7#2),g=3,random(2,4,5),g=4,random(3,5),g=5, random(2,3,4),g=6,random(5,7),g=7,random(2,3,4),g=8,random(3,5,7),g=9,random(2,4,5),g=10,random(3,7),g=11,random(2..5))", "name": "f", "description": ""}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..7)", "name": "a", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(a=1, random(2..7),a=2,random(3..7#2),a=3,random(4,5,7),a=4,random(5,7),a=5, random(6,7,8),a=6,random(7,11),a=7,random(8,9))", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "b", "g", "f", "s1", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Write the following expression as a single fraction in its lowest form:
", "tags": ["checked2015", "Fractions", "fractions", "lowest form", "mas1601", "MAS1601", "simplifying fractions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded description.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\frac{a} {b + \\frac{c}{d}}$ as a single fraction in the form $\\displaystyle \\frac{p}{q}$ for integers $p$ and $q$.
"}, "advice": "We have:
\\[\\simplify[std]{{g} / ({a} + {s1} * ({b * g} / {f})) = {g} / (({a} * {f} + {s1} * {b * g}) / {f}) ={g} / (({a * f + s1 * b * g}) / {f})= ({f}*{g}) / ({a * f + s1 * b * g}) = ({g * f} / {(a * f + s1 * b * g)})}\\]
Here we use the result that dividing by a fraction $\\frac{a}{b}$ is the same as multiplying by $\\frac{b}{a}$.
The resulting fraction is in lowest form i.e. the top and bottom do not have a common factor.
Input as a fraction or an integer, not as a decimal.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n\\[\\simplify{{a} * x + {b} = {c} * x + {d}}\\]
\n$x=\\;$ [[0]]
\n \n ", "marks": 0}], "statement": "\nSolve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n \n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "Solving equations", "subject of an equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle ax+b = cx+d$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "If $\\simplify{{a}*x + {b}}=\\simplify{{c}*x + {d}}$, then we first subtract $\\var{c}x$ from each side.
\nWe obtain $\\simplify{{a}x - {c}x+ {b}}= \\var{d}$.
\nNext we subtract $\\var{b}$ from each side (or add $\\var{-b}$ if you prefer - it means the same thing), to obtain $\\var{a}x - \\var{c}x = \\simplify{{d}-{b}}$.
\nIn other words $\\simplify{{a}-{c}}x= \\simplify{{d}-{b}}$.
\nNow divide both sides by $\\simplify{{a-c}}$ to obtain $x=\\dfrac{\\simplify{{d-b}}}{\\simplify{{a-c}}}$ $=\\Large{\\simplify{{an1}/{an2}}}$.
"}, {"name": "Solve a pair of simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "aort": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b*b1>0,'take away the equation','add the equation')", "name": "aort", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sb*random(1..9)", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sc*random(1..9)", "name": "c", "description": ""}, "sc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "sc", "description": ""}, "sc1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "sc1", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a*b2=a1*b,b2+1,b2)", "name": "b1", "description": ""}, "fromorto": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b*b1>0,'from','to')", "name": "fromorto", "description": ""}, "that": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(abs(b),abs(b1))/abs(b1)", "name": "that", "description": ""}, "sb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "sb", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "b2", "description": ""}, "this": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(abs(b),abs(b1))/abs(b)", "name": "this", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sc1*random(1..9)", "name": "c1", "description": ""}, "s6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b*b1>0,-1,1)", "name": "s6", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sa*random(2..9)", "name": "a", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(a2=2,random(3,5,7,9),a2=3,random(2,4,5,7),a2=4,random(3,5,7,9),a2=5,random(3,4,6,7,9),a2=6,random(4,5,7,8,9),a2=7,random(3,4,5,6,8,9),a2=8,random(3,5,6,7,9),a2=9,random(2,4,5,7,8),9)", "name": "a1", "description": ""}, "sa": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "sa", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(a)", "name": "a2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "that", "this", "sc1", "s1", "s6", "a1", "aort", "a2", "b1", "b2", "sc", "sb", "sa", "fromorto", "c1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "{c*b1-b*c1}/{b1*a-a1*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}, {"answer": "{c*a1-a*c1}/{b*a1-a*b1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "prompt": "", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "prompt": "\n\t\t\t\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray} \\]
\n\t\t\t$x=\\phantom{{}}$[[0]], $y=\\phantom{{}}$[[1]]
\n\t\t\tInput your answers as fractions or integers, not as decimals.
\n\t\t\tSee \"Show steps\" for a video that describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.
\n\t\t\t \n\t\t\t", "stepsPenalty": 0}], "statement": "Solve the following simultaneous equations for $x$ and $y$. Input your answers as fractions or integers, not as decimals.
", "tags": ["ACC1012", "checked2015", "equations", "linear", "pair of linear equations", "simultaneous", "simultaneous linear equations", "solve linear equations", "solving equations", "Solving equations", "video"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t \t\t \t\t \t\t \t\t \t\t \t\t5/08/2012:
\n\t\t \t\t \t\t \t\t \t\t \t\t \t\tAdded more tags.
\n\t\t \t\t \t\t \t\t \t\t \t\t \t\tAdded description.
\n\t\t \t\t \t\t \t\t \t\t \t\t \t\tChecked calculation. OK.
\n\t\t \t\t \t\t \t\t \t\t \t\t \n\t\t \t\t \t\t \t\t \t\t \n\t\t \t\t \t\t \t\t \n\t\t \t\t \t\t \n\t\t \t\t \n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$ and $y$: \\[ \\begin{eqnarray} a_1x+b_1y&=&c_1\\\\ a_2x+b_2y&=&c_2 \\end{eqnarray} \\]
\nThe included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}&\\mbox{ ........(1)}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1}&\\mbox{ ........(2)} \\end{eqnarray} \\]
To get a solution for $x$ multiply equation (1) by {this} and equation (2) by {that}
This gives:
\\[ \\begin{eqnarray} \\simplify[std]{{a*this}x+{b*this}y}&=&\\var{this*c}&\\mbox{ ........(3)}\\\\ \\simplify[std]{{a1*that}x+{b1*that}y}&=&\\var{that*c1}&\\mbox{ ........(4)} \\end{eqnarray} \\]
Now {aort} (4) {fromorto} equation (3) to get
\\[\\simplify[std]{({a*this}+{s6*a1*that})x={this*c}+{s6*that*c1}}\\]
And so we get the solution for $x$:
\\[x = \\simplify{{c*b1-b*c1}/{b1*a-a1*b}}\\]
Substituting this value into any of the equations (1) and (2) gives:
\\[y = \\simplify{{c*a1-a*c1}/{b*a1-a*b1}}\\]
You can check that these solutions are correct by seeing if they satisfy both equations (1) and (2) by substituting these values into the equations.
Input as a fraction or an integer, not as a decimal.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[\\simplify{({p}*x+{s}) / ({a} * x + {b}) = ({q}*x+{t}) / ({c} * x + {d})}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "steps": [{"type": "information", "prompt": "\nCross-multiply to get:
\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]
Multiplying out to get \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}.\\] Subtract the $x^2$ term from each side to leave a linear equation:
Solve this equation for $x$.
\n \n ", "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "showCorrectAnswer": true, "stepsPenalty": 1}], "statement": "\nSolve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n \n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "subject of an equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{px+s}{ax+b} = \\frac{qx+t}{cx+d}$ with $pc=qa$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Cross-multiply to get:
\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]
Multiplying out we get \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}\\] Subtracting ${\\var{a*q}}x^2$ from each side we are left with \\[\\simplify{{p*d+c*s}x+{s*d}={q*b+t*a}x+{t*b}}\\] which we solve as a linear equation: \\[\\simplify{{p*d+c*s-q*b-t*a}x={t*b-s*d}}\\] and so \\[\\simplify{x={an1}/{an2}}.\\]
Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "\nRearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now solve this linear equation.
\\[\\simplify{{s} / ({a} * x + {b}) = {t} / ({c} * x + {d})}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "stepsPenalty": 1}], "statement": "\nSolve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "subject of an equation"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{s}{ax+b} = \\frac{t}{cx+d}$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Rearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now this is a linear equation which is solved in the following steps: \\[\\simplify{{s*c-t*a}*x={t*b-s*d}}\\] and then \\[\\simplify{x={t*b-s*d}/{s*c-t*a}}.\\]
Input your answer as a fraction or an integer. Do not input the answer as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[\\simplify[std]{{a} * x + {b} = {c} * x + {d}}\\]
Input your answer as a fraction or an integer. Do NOT input the answer as a decimal.
$x\\;=$[[0]]
", "marks": 0}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..12)", "name": "b", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b=td,td+1,td)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..12)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a=tc,tc+1,tc)", "name": "c", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s3", "description": ""}, "tc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..20)", "name": "tc", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "td": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..20)", "name": "td", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "td", "tc"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Solve the following linear equation for $x$.
", "tags": ["checked2015", "equations", "linear equation", "mas1601", "MAS1601", "solving equations", "Solving equations", "solving linear equations"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded more tags.
\nAdded description.
\nChecked calculation. OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve $\\displaystyle ax + b = cx + d$ for $x$.
"}, "advice": "Given the equation \\[\\simplify[std]{{a}x+{b}={c}x+{d}}\\] we first collect together all the constant terms, and collect together all the terms in $x$.
\nThe equation can then be written as:
\\[\\simplify[std]{({a}-{c})x=({d}+{-b})}\\] i.e.
\\[\\simplify{{a-c}x={d-b}}\\]
which gives \\[x =\\simplify[std]{{(d-b)}/{(a-c)}}\\] as the solution.
Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{c*a1-a*c1}/{b*a1-a*b1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray} \\]
\n$x=\\phantom{{}}$[[0]], $y=\\phantom{{}}$[[1]]
\nInput your answers as fractions or integers, not as decimals.
", "marks": 0}], "statement": "Solve the following simultaneous equations for $x$ and $y$. Input your answers as fractions or integers, not as decimals.
", "tags": ["checked2015", "equations", "linear", "mas1601", "MAS1601", "pair of linear equations", "simultaneous", "simultaneous linear equations", "solve linear equations", "solving equations", "Solving equations"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded more tags.
\nAdded description.
\nChecked calculation. OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$ and $y$: \\[ \\begin{eqnarray} a_1x+b_1y&=&c_1\\\\ a_2x+b_2y&=&c_2 \\end{eqnarray} \\]
"}, "advice": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}&\\mbox{ ........(1)}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1}&\\mbox{ ........(2)} \\end{eqnarray} \\]
To get a solution for $x$ multiply equation (1) by {this} and equation (2) by {that}
This gives:
\\[ \\begin{eqnarray} \\simplify[std]{{a*this}x+{b*this}y}&=&\\var{this*c}&\\mbox{ ........(3)}\\\\ \\simplify[std]{{a1*that}x+{b1*that}y}&=&\\var{that*c1}&\\mbox{ ........(4)} \\end{eqnarray} \\]
Now {aort} (4) {fromorto} equation (3) to get
\\[\\simplify[std]{({a*this}+{s6*a1*that})x={this*c}+{s6*that*c1}}\\]
And so we get the solution for $x$:
\\[x = \\simplify{{c*b1-b*c1}/{b1*a-a1*b}}\\]
Substituting this value into any of the equations (1) and (2) gives:
\\[y = \\simplify{{c*a1-a*c1}/{b*a1-a*b1}}\\]
You can check that these solutions are correct by seeing if they satisfy both equations (1) and (2) by substituting these values into the equations.
$-\\var{a}-n$
", "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "-({t}^2)", "vsetrange": [0, 1], "checkingaccuracy": 0, "checkvariablenames": false, "expectedvariablenames": [], "prompt": "$-(n^2)$
", "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{q}", "vsetrange": [0, 1], "checkingaccuracy": 0, "checkvariablenames": false, "expectedvariablenames": [], "prompt": "$n^3-\\var{b}n$
", "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "statement": "\nFind the value of the following expressions, given the stated value of $n$.
\n$n=-\\var{t}$
\n \n \n \n \n \n ", "tags": ["ACC1012", "checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "What is the value of the expression given a choice of n?
"}, "advice": ""}, {"name": "Combining algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "name": "s1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "description": "", "name": "b2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,-a])", "description": "", "name": "c"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "name": "nb"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,round(a*b/a1)])", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "b1", "b2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2+a1*c}*x^2 + {b*c+a1*b2+b1*a2+a*d} * x + {b1 * d + b2 * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "musthave": {"message": "Input as a single fraction with the numerator as a quadratic and all terms expanded in the numerator.
", "showStrings": false, "partialCredit": 0, "strings": ["^"]}, "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "variableReplacements": [], "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\nNote: you do not need to expand the denominator, but you must enter the numerator as a polynomial in $x$.
\nInput the fraction here: [[0]]
\nClick on Show steps for more information. You will lose one mark if you do so.
", "steps": [{"prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / (b*d)}\\]
and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nNote that in your answer you do not need to expand the denominator.
", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 1}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t \t\t5/08/2012:
\n \t\t \t\t \t\tAdded tags.
\n \t\t \t\t \t\tAdded description.
\n \t\t \t\t \t\tChanged to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t \t\t12/08/2012:
\n \t\t \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t \t\tChecked calculation.OK.
\n \t\t \t\t \t\tImproved display in content areas.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "let(\n qa,a*a2+a1*c,\n qb,b*c+a1*b2+b1*a2+a*d,\n qc,b1*d+b2*b,\n roots,[-b/a1,-d/a2],\n \n not (((-qb+sqrt(qb*qb+4*qa*qc))/(2*qa) in roots) or ((-qb-sqrt(qb*qb+4*qa*qc))/(2*qa) in roots))\n)", "maxRuns": "300"}, "advice": "The formula for {nb} fractions is :
\n\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / b*d}\\]
\nand for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nHence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
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.
\nThe shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.
\nAuthor 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.
a)
\nThe area of a rectangle is calculated using the formula
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}\\text{.}\\]
\nWe have a base of $\\var{w0}$m and a height $\\var{h0}$m, therefore
\n\\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}
b)
\nThe parallelogram is just a slanted rectangle:
\n\n
Therefore, the area of a parallelogram is calculated using the formula
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]
\nWe have a base $\\var{w1}$m and perpendicular height $\\var{h1}$m.
\n\\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}
The area of a triangle is calculated using the formula
\n\\[\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\\]
\nNote that the triangle is half of a rectangle:
\n\nOur triangle has a base $\\var{w2}$m and a height $\\var{h2}$m, therefore
\n\\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}
d)
\n\nA trapezium can be interpreted as half of a parallelogram, this is shown below:
\n\nAs we only want the area of one half of this shape, the area is half of
\n\\[\\mathrm{area} = (a+b) \\times \\mathrm{height}\\text{,}\\]
\nwith ${a} = \\var{w5a}$m, ${b} = \\var{w5b}$m, and height $\\var{h5}$m.
\n\\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}
Calculate the area of the following shapes.
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Numbas.exam.currentQuestion.parts[3].gaps[1].display.studentAnswer(d.Y());\n});\ne.on('drag',function(){\n Numbas.exam.currentQuestion.parts[4].gaps[0].display.studentAnswer(e.X());\n Numbas.exam.currentQuestion.parts[4].gaps[1].display.studentAnswer(e.Y());\n});\nf.on('drag',function(){\n Numbas.exam.currentQuestion.parts[5].gaps[0].display.studentAnswer(f.X());\n Numbas.exam.currentQuestion.parts[5].gaps[1].display.studentAnswer(f.Y());\n});\n\n/*\nquestion.signals.on('HTMLAttached',function(e) {\n ko.computed(function(){ \n var x = parseFloat(question.parts[0].gaps[0].display.studentAnswer());\n var y = parseFloat(question.parts[0].gaps[1].display.studentAnswer());\n if(!(isNaN(x) || isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n a.moveTo([x,y],100);\n }\n });\n});\n*/\n\nreturn div;\n\n"}}, "advice": "Coordinates are given as $(x\\text{-coordinate},y\\text{-coordinate})$. So, for example $(2,5)$ has $x$-coordinate $= 2$ and $y$-coordinate$= 5$.
\nPlot the first coordinate ($x$-coordinate) against the horizontal axis and then plot the second coordinate ($y$-coordinate) against the vertical axis.
\n{correctPoints()}
"}, {"name": "Mohammad's copy of Mathematical formulae - Volume", "extensions": [], "custom_part_types": [], "resources": [["question-resources/sqbasedpyramid_sEpkGzO.svg", "/srv/numbas/media/question-resources/sqbasedpyramid_sEpkGzO.svg"], ["question-resources/triangularprism.svg", "/srv/numbas/media/question-resources/triangularprism.svg"], ["question-resources/cylinder.svg", "/srv/numbas/media/question-resources/cylinder.svg"], ["question-resources/cuboid.svg", "/srv/numbas/media/question-resources/cuboid.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/"}, {"name": "Mohammad Rahman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18139/"}], "tags": ["3D shapes", "cuboid", "Cylinder", "cylinder", "pyramid", "taxonomy", "triangular prism", "Volume", "volume", "volume of a cuboid", "volume of a cylinder", "volume of a pyramid", "volume of a triangular prism"], "metadata": {"description": "Calculate the volume of different 3D shapes, given the units and measurements required. The formulae for the volume of each shape are available as steps if required.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Calculate the volumes of the following shapes.
", "advice": "a)
\nFor a cuboid, we first need to find out the area of one of the faces then multiply this area by the depth of the object.
In this example you can choose either of the faces. To make the calculations easier I am going to choose the face with $\\mathrm{base} = \\var{d4}m$ and $\\mathrm{height}= \\var{h4}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\square} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{h4} \\times \\var{d4} \\\\
&= \\var{h4*d4}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the face ($\\mathrm{Area\\thinspace_\\square}$) we can multiply this by the $\\mathrm{depth} = \\var{w4}m$ to calculate the volume of the object.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\var{h4*d4} \\times \\var{w4} \\\\
&= \\var{h4*d4*w4}\\, \\mathrm{m}^3\\,.
\\end{align}
b)
\nFor a triangular prism, we first need to find the area of one of the faces then multiply this area by the depth of the prism.
In this example the easiest way to calculate the volume is to take the area of the triangular face first with $\\mathrm{base} = \\var{w6}m$ and $\\mathrm{height} = \\var{h6}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\triangle} &= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\\\
&= \\frac{\\var{w6} \\times \\var{h6}}{2} \\\\
&= \\var{0.5*w6*h6}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the triangular face ($\\mathrm{Area\\thinspace_\\triangle}$) we can multiply this by the $\\mathrm{depth} = \\var{d6}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\var{w6*h6} \\times \\var{d6} \\\\
&= \\var{w6*h6*d6}\\, \\mathrm{m}^2\\,.
\\end{align}
c)
\nFor a cylinder, we first need to find the area of the circular face then multiply this area by the depth of the cylinder.
In this example the radius of the circular face is $\\mathrm{radius} = \\var{r7}m$ which can be used to calculate the area of the circular face.
\\begin{align}
\\mathrm{Area\\thinspace_\\bigcirc} &= \\pi \\times \\mathrm{radius}^2 \\\\
&= \\pi \\times \\var{r7}^2 \\\\
&= \\var{pi * (r7)^2}\\, \\mathrm{m}^2 \\,.
\\end{align}
Now that we have the area of the circular face ($\\mathrm{Area\\thinspace_\\bigcirc}$) we can multiply this by the $\\mathrm{depth} =\\var{w7}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\var{pi*(r7)^2} \\times \\var{w7} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 5)} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 1)}\\, \\mathrm{m}^2\\,. \\quad \\text{1 d.p.}
\\end{align}
d)
For a rectangular-based pyramid, we first need to calculate the area of the base and multiply this area by $\\frac{1}{3}$ the height of the pyramid.
In this example the area of the base can be calculated from the $\\mathrm{width}= \\var{w8}m$ and $\\mathrm{length} = \\var{d8}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\boxdot} &= \\mathrm{width} \\times \\mathrm{length} \\\\
&= \\var{w8} \\times \\var{d8} \\\\
&= \\var{w8*d8}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the base we can multiply this by the $\\frac{1}{3} \\mathrm{height}$ where $\\mathrm{height} = \\var{h8}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3} \\mathrm{height} \\\\
&= \\var{w8*d8} \\times \\var{dpformat(1/3*h8,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,1)}\\, \\mathrm{m}^3\\,. \\quad \\text{1 d.p.}
\\end{align}
Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w6": {"name": "w6", "group": "Triangular prism", "definition": "random(5..9#1)", "description": "Creates base of triangle.
", "templateType": "anything", "can_override": false}, "d8": {"name": "d8", "group": "Square based pyramid", "definition": "random(3..6#0.1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h8": {"name": "h8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "Height of pyramid.
", "templateType": "anything", "can_override": false}, "w7": {"name": "w7", "group": "Cylinder", "definition": "random(7..15#0.1)", "description": "Depth of cylinder.
", "templateType": "anything", "can_override": false}, "d6": {"name": "d6", "group": "Triangular prism", "definition": "random(9..15#0.1)", "description": "Depth of triangular prism.
", "templateType": "anything", "can_override": false}, "r7": {"name": "r7", "group": "Cylinder", "definition": "random(2..6#1)", "description": "Radius of the cylinder.
", "templateType": "anything", "can_override": false}, "h4": {"name": "h4", "group": "Cuboid ", "definition": "random(2..5#1 except d4)", "description": "Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w4": {"name": "w4", "group": "Cuboid ", "definition": "random(5.5..8#0.1)", "description": "Width of cuboid.
", "templateType": "anything", "can_override": false}, "w8": {"name": "w8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h6": {"name": "h6", "group": "Triangular prism", "definition": "random(2..5#1)", "description": "Height of traingle.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Cuboid ", "variables": ["w4", "d4", "h4"]}, {"name": "Triangular prism", "variables": ["w6", "h6", "d6"]}, {"name": "Cylinder", "variables": ["r7", "w7"]}, {"name": "Square based pyramid", "variables": ["h8", "w8", "d8"]}], "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 $\\mathrm{Volume}$ of the following cuboid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cuboid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\mathrm{base} \\times \\mathrm{height} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following triangular prism.
\n\n$\\mathrm{Volume} =$[[0]]$\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a triangular prism:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following cylinder.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cylinder:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\pi \\times \\mathrm{r}^2 \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following pyramid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a square-based pyramid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3}\\mathrm{height} \\\\
&= \\mathrm{width} \\times \\mathrm{length} \\times \\frac{1}{3}\\mathrm{height}
\\end{align}
Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "prompt": "The equation of the line is of the form $y=mx+c$.
\nYou are given the gradient $m$ and you can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Find the equation of the straight line which:
\n\n
\n
Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\nInput $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\nClick on Show steps if you need help, you will lose 1 mark if you do so.
", "tags": ["checked2015", "diagram", "equation of a straight line", "gradient of a line", "mas1601", "MAS1601", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
\nImproved display in content areas. Corrected some minor typos.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The equation of the line is of the form $y=mx+c$.
\nYou are given the gradient $\\displaystyle m= \\simplify{{b-d}/{a-c}}$ and we can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
\nUsing this we get:
\\[ \\begin{eqnarray} \\var{b}&=&\\simplify[std]{({b-d}/{a-c}){a}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{b}-({b-d}/{a-c}){a}={(b*c-a*d)}/{(c-a)}} \\end{eqnarray} \\]
Hence the equation of the line is
\\[y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}\\]
{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.
\nWhat is the engine size in litres?
\n[[0]] litres.
\n"}, {"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.
\nWhat is the engine size of the first car in units of m$^3$?
\n[[0]]m$^3$ Give your answer to 4 decimal places
"}], "advice": "The advertised engine size is $\\var{cc}$ cubic centimetres. To convert cubic centimetres to litres, we divide by $1000$.
\n\\[\\var{cc}\\div 1000= \\var{litres}\\text{ litres.}\\]
\nIn order to convert to cubic metres, we first note that
\n\\[ 1 \\text{cm} = 0.01 \\text{m.} \\]
\nAn example of a volume of $1\\text{cm}^3$ is a cube with $1$cm sides. Converting each side into metres,
\n\\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
\n\\[ \\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.
\n{person['name']} recognises that 'cc' stands for units of cubic centimetres (cm$^3$) and knows the following conversions:
\n| $1$ m | \n$100$ cm | \n
| $1$ litre | \n$1000\\text{cm}^3$ | \n
Convert figures for car engine sizes between cc (cm^3), litres, and m^3.
"}}, {"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": "The price per hour is the total price divide by the number of hours.
\n\\[ \\text{Price per hour} = \\frac{\\var{block_price_per_hour*hours}}{\\var{hours}} = £\\var{dpformat(block_price_per_hour,2)} \\text{ per hour} \\]
\nThe price is given in pence per minute. To convert to pounds per minute, divide by $100$:
\n\\[ \\var{100*competitor_price_per_minute} \\text{ p/minute} = £\\var{dpformat(competitor_price_per_minute,2)} \\text{ per minute} \\]
\nThen to convert to pounds per hour, multiply by $60$:
\n\\[ £\\var{dpformat(competitor_price_per_minute,2)} \\text{ per minute} = £\\var{dpformat(competitor_price_per_minute*60,2)} \\text{ per hour} \\]
\n{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.
\nTo compare the two prices, {pronouns['they']} decide{verbs} to convert the advance booking price to a price per hour.
\nPrice 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.
\nTo compare this price to ACME's offer, {pname} decide{verbs} to convert it to a price per hour.
\nPrice per hour: £ [[0]]
\n", "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 density given mass and volume", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "advice": "We are told that the ball has a volume of $\\var{volume}\\text{cm}^3$ and a mass of $\\var{mass}$g, and we are asked to calculate the density of the box in g/cm$^3$.
\nThe formula for density is
\n\\[\\begin{align} \\text{Density} &= \\frac{\\text{Mass}}{\\text{Volume}} \\\\[4pt]
&= \\frac{\\var{mass}}{\\var{volume}} \\\\[4pt]
&= \\var{density} \\\\
&= \\var{precround(density,2)}\\text{g/cm}^3\\text{.} \\\\
\\end{align}\\]
Since the density of the ball is {if(density>1,'greater','smaller')} than the density of water, {person['name']}'s ball will {if(density>1,'sink','float')}.
\n", "statement": "A solid object placed in water will sink if its density is greater than that of water ($1\\text{g/cm}^3$).
\n{person['name']}'s toy ball has a volume of $\\var{volume}\\text{cm}^3$ and a mass of $\\var{mass}$g. Whilst playing, {person['pronouns']['they']} drops {person['pronouns']['their']} ball into a pond.
", "variables": {"mass": {"name": "mass", "group": "Ungrouped variables", "definition": "random(55..65)", "templateType": "anything", "description": "mass of the box
"}, "person": {"name": "person", "group": "Ungrouped variables", "definition": "random_person()", "templateType": "anything", "description": ""}, "density": {"name": "density", "group": "Ungrouped variables", "definition": "mass/volume", "templateType": "anything", "description": ""}, "volume": {"name": "volume", "group": "Ungrouped variables", "definition": "random(40..70 except mass)", "templateType": "anything", "description": "Volume of box
"}, "mark_matrix": {"name": "mark_matrix", "group": "Ungrouped variables", "definition": "[if(density<1,1,0),if(density>1,1,0)]", "templateType": "anything", "description": ""}}, "tags": ["calculating density", "compound units", "Compound units", "density", "mass", "taxonomy", "Volume", "volume"], "ungrouped_variables": ["volume", "mass", "density", "person", "mark_matrix"], "functions": {}, "metadata": {"description": "Calculate the density of an object given its mass and volume.
", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "showCorrectAnswer": true, "marks": 0, "prompt": "The relationship between density, mass and volume is
\n\\[\\text{Density} = \\frac{\\text{Mass}}{\\text{Volume}}.\\]
", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "precisionPartialCredit": 0, "strictPrecision": false, "correctAnswerFraction": false, "minValue": "density", "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "density", "showCorrectAnswer": true, "precision": "2", "type": "numberentry", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "variableReplacementStrategy": "originalfirst", "precisionMessage": "Round your answer to $3$ significant figures.
"}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "prompt": "What is the density of the ball?
\n[[0]] g/cm$^3$. Round your answer to $2$ decimal places.
", "stepsPenalty": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "maxMarks": 0, "displayType": "radiogroup", "choices": ["It floats
", "It sinks
"], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "shuffleChoices": false, "prompt": "Does {person['name']}'s ball float or sink?
", "marks": 0, "showFeedbackIcon": true, "matrix": "mark_matrix", "variableReplacementStrategy": "originalfirst", "minMarks": 0}], "type": "question", "variable_groups": [], "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}}, {"name": "Mohammad's copy of Mathematical formulae - Volume", "extensions": [], "custom_part_types": [], "resources": [["question-resources/sqbasedpyramid_sEpkGzO.svg", "/srv/numbas/media/question-resources/sqbasedpyramid_sEpkGzO.svg"], ["question-resources/triangularprism.svg", "/srv/numbas/media/question-resources/triangularprism.svg"], ["question-resources/cylinder.svg", "/srv/numbas/media/question-resources/cylinder.svg"], ["question-resources/cuboid.svg", "/srv/numbas/media/question-resources/cuboid.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/"}, {"name": "Mohammad Rahman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18139/"}], "tags": ["3D shapes", "cuboid", "Cylinder", "cylinder", "pyramid", "taxonomy", "triangular prism", "Volume", "volume", "volume of a cuboid", "volume of a cylinder", "volume of a pyramid", "volume of a triangular prism"], "metadata": {"description": "Calculate the volume of different 3D shapes, given the units and measurements required. The formulae for the volume of each shape are available as steps if required.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Calculate the volumes of the following shapes.
", "advice": "a)
\nFor a cuboid, we first need to find out the area of one of the faces then multiply this area by the depth of the object.
In this example you can choose either of the faces. To make the calculations easier I am going to choose the face with $\\mathrm{base} = \\var{d4}m$ and $\\mathrm{height}= \\var{h4}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\square} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{h4} \\times \\var{d4} \\\\
&= \\var{h4*d4}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the face ($\\mathrm{Area\\thinspace_\\square}$) we can multiply this by the $\\mathrm{depth} = \\var{w4}m$ to calculate the volume of the object.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\var{h4*d4} \\times \\var{w4} \\\\
&= \\var{h4*d4*w4}\\, \\mathrm{m}^3\\,.
\\end{align}
b)
\nFor a triangular prism, we first need to find the area of one of the faces then multiply this area by the depth of the prism.
In this example the easiest way to calculate the volume is to take the area of the triangular face first with $\\mathrm{base} = \\var{w6}m$ and $\\mathrm{height} = \\var{h6}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\triangle} &= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\\\
&= \\frac{\\var{w6} \\times \\var{h6}}{2} \\\\
&= \\var{0.5*w6*h6}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the triangular face ($\\mathrm{Area\\thinspace_\\triangle}$) we can multiply this by the $\\mathrm{depth} = \\var{d6}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\var{w6*h6} \\times \\var{d6} \\\\
&= \\var{w6*h6*d6}\\, \\mathrm{m}^2\\,.
\\end{align}
c)
\nFor a cylinder, we first need to find the area of the circular face then multiply this area by the depth of the cylinder.
In this example the radius of the circular face is $\\mathrm{radius} = \\var{r7}m$ which can be used to calculate the area of the circular face.
\\begin{align}
\\mathrm{Area\\thinspace_\\bigcirc} &= \\pi \\times \\mathrm{radius}^2 \\\\
&= \\pi \\times \\var{r7}^2 \\\\
&= \\var{pi * (r7)^2}\\, \\mathrm{m}^2 \\,.
\\end{align}
Now that we have the area of the circular face ($\\mathrm{Area\\thinspace_\\bigcirc}$) we can multiply this by the $\\mathrm{depth} =\\var{w7}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\var{pi*(r7)^2} \\times \\var{w7} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 5)} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 1)}\\, \\mathrm{m}^2\\,. \\quad \\text{1 d.p.}
\\end{align}
d)
For a rectangular-based pyramid, we first need to calculate the area of the base and multiply this area by $\\frac{1}{3}$ the height of the pyramid.
In this example the area of the base can be calculated from the $\\mathrm{width}= \\var{w8}m$ and $\\mathrm{length} = \\var{d8}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\boxdot} &= \\mathrm{width} \\times \\mathrm{length} \\\\
&= \\var{w8} \\times \\var{d8} \\\\
&= \\var{w8*d8}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the base we can multiply this by the $\\frac{1}{3} \\mathrm{height}$ where $\\mathrm{height} = \\var{h8}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3} \\mathrm{height} \\\\
&= \\var{w8*d8} \\times \\var{dpformat(1/3*h8,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,1)}\\, \\mathrm{m}^3\\,. \\quad \\text{1 d.p.}
\\end{align}
Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w6": {"name": "w6", "group": "Triangular prism", "definition": "random(5..9#1)", "description": "Creates base of triangle.
", "templateType": "anything", "can_override": false}, "d8": {"name": "d8", "group": "Square based pyramid", "definition": "random(3..6#0.1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h8": {"name": "h8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "Height of pyramid.
", "templateType": "anything", "can_override": false}, "w7": {"name": "w7", "group": "Cylinder", "definition": "random(7..15#0.1)", "description": "Depth of cylinder.
", "templateType": "anything", "can_override": false}, "d6": {"name": "d6", "group": "Triangular prism", "definition": "random(9..15#0.1)", "description": "Depth of triangular prism.
", "templateType": "anything", "can_override": false}, "r7": {"name": "r7", "group": "Cylinder", "definition": "random(2..6#1)", "description": "Radius of the cylinder.
", "templateType": "anything", "can_override": false}, "h4": {"name": "h4", "group": "Cuboid ", "definition": "random(2..5#1 except d4)", "description": "Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w4": {"name": "w4", "group": "Cuboid ", "definition": "random(5.5..8#0.1)", "description": "Width of cuboid.
", "templateType": "anything", "can_override": false}, "w8": {"name": "w8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h6": {"name": "h6", "group": "Triangular prism", "definition": "random(2..5#1)", "description": "Height of traingle.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Cuboid ", "variables": ["w4", "d4", "h4"]}, {"name": "Triangular prism", "variables": ["w6", "h6", "d6"]}, {"name": "Cylinder", "variables": ["r7", "w7"]}, {"name": "Square based pyramid", "variables": ["h8", "w8", "d8"]}], "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 $\\mathrm{Volume}$ of the following cuboid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cuboid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\mathrm{base} \\times \\mathrm{height} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following triangular prism.
\n\n$\\mathrm{Volume} =$[[0]]$\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a triangular prism:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following cylinder.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cylinder:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\pi \\times \\mathrm{r}^2 \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following pyramid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a square-based pyramid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3}\\mathrm{height} \\\\
&= \\mathrm{width} \\times \\mathrm{length} \\times \\frac{1}{3}\\mathrm{height}
\\end{align}
Find the original price before a discount by dividing the new price by the percentage discount.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{name1} and {name2} are friends. {name1} noticed {name2}'s new {item} when he came over to visit her house. He immediately knew he wanted to buy the same model. When he got home, he bought the {item} online for £{newprice}.
", "advice": "We need to find the original price paid by {name2}. This value represents 100%.
\nBy the time {name1} bought the {item}, the price had decreased by {percentage}%.
\n{name1} therefore paid {100-percentage}% of the price {name2} paid.
\n\nWe use the unitary method to find the original price. We know the price paid by {name1}.
\n\\[\\var{100-percentage}\\text{%} = \\var{newprice} \\text{.}\\]
\nDivide both sides by {100-percentage} to get
\n\\[\\begin{align} 1\\text{%} &= \\var{newprice} \\div \\var{100-percentage} \\\\&= \\var{newprice/(100-percentage)} \\text{.} \\end{align}\\]
\nMultiply both sides by 100 to get
\n\\[\\begin{align} 100\\text{%} &= \\var{newprice/(100-percentage)} \\times 100 \\\\&= \\var{newprice/(100-percentage)*100} \\\\&= \\var{oldprice}\\text{.} \\end{align}\\]
\nThis is the original price paid by {name2} before the {percentage}% decrease.
\nWe can check our answer with a different method.
\n\\[\\begin{align} \\var{100-percentage}\\text{% of } \\var{oldprice} &= \\var{(100-percentage)/100} \\times \\var{oldprice} \\\\&= \\var{(100-percentage)/100*oldprice} \\\\&= \\var{precround((100-percentage)/100*oldprice, 2)} \\text{.} \\end{align}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"newprice": {"name": "newprice", "group": "Ungrouped variables", "definition": "precround(oldprice*(100-percentage)/100,2)", "description": "", "templateType": "anything", "can_override": false}, "name2": {"name": "name2", "group": "Ungrouped variables", "definition": "random(\"Emily\", \"Kate\", \"Michaela\", \"Susan\", \"Sophie\")", "description": "A female name.
", "templateType": "anything", "can_override": false}, "name1": {"name": "name1", "group": "Ungrouped variables", "definition": "random(\"Josh\", \"Adam\", \"Mike\", \"Trevor\", \"Alex\")", "description": "A male name.
", "templateType": "anything", "can_override": false}, "oldprice": {"name": "oldprice", "group": "Ungrouped variables", "definition": "switch(\n item = \"TV\", random(179.99..1199.99 #10), \n item = \"laptop\", random(209.99..799.99 #10),\n item = \"smartphone\", random(109.99..799.99 #10),\n item = \"PC\", random(209.99..969.99 #10),\n item = \"gaming console\", random(89.99..349.99 #10),\n 399.99)", "description": "", "templateType": "anything", "can_override": false}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "random(5..30)", "description": "Discount percentage.
", "templateType": "anything", "can_override": false}, "item": {"name": "item", "group": "Ungrouped variables", "definition": "random(\"TV\", \"laptop\", \"smartphone\", \"PC\", \"gaming console\")", "description": "The bought item.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "precround(precround(oldprice*(100-percentage)/100,2)*100/(100-percentage),2) = oldprice", "maxRuns": "1000"}, "ungrouped_variables": ["item", "name1", "percentage", "name2", "oldprice", "newprice"], "variable_groups": [], "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": "When {name1} told {name2} how much he had paid for the {item}, {name2} said the price had decreased by {percentage}% since she bought it.
\nHow much did {name2} pay for the {item}?
\n£ [[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "oldprice", "maxValue": "oldprice", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Calculating a simple rate of pay", "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", "rate of pay", "taxonomy"], "metadata": {"description": "Calculate a rate of pay (in pounds per week) given the total pay over a given period of time.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{pname} gets paid a total of $£\\var{payment}$ at the end of {their} summer job; {they} work{s} for $\\var{weeks}$ weeks.
", "advice": "We are told that {pname} gets paid a total of $£\\var{payment}$ at the end of {their} summer job and that {they} work{s} at {their} job for $\\var{weeks}$ weeks.
\nTo calculate the amount of money {pname} gets paid per week, we divide the total amount of money that {they} earn{s} at the end of {their} job by how many weeks that {they} work{s} for.
\n\\[£\\displaystyle\\frac{\\var{payment}}{\\var{weeks}} = £\\var{{payment/weeks}}.\\]
\nTherefore {pname} gets paid $£\\var{{payment/weeks}}$/week.
\nNote that in compound measures, a forward slash symbol / is often used instead of the word 'per'. So $£\\var{{payment/weeks}}$/week means the same as $£\\var{{payment/weeks}}$ per week.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"person": {"name": "person", "group": "A person", "definition": "random_person()", "description": "A random person
", "templateType": "anything", "can_override": false}, "weeks": {"name": "weeks", "group": "Ungrouped variables", "definition": "random(5,8)", "description": "Number of weeks person works for
", "templateType": "anything", "can_override": false}, "payment": {"name": "payment", "group": "Ungrouped variables", "definition": "random(1400,1600,1800)", "description": "amount person gets paid
", "templateType": "anything", "can_override": false}, "pname": {"name": "pname", "group": "A person", "definition": "person[\"name\"]", "description": "", "templateType": "anything", "can_override": false}, "they": {"name": "they", "group": "A person", "definition": "person[\"pronouns\"][\"they\"]", "description": "", "templateType": "anything", "can_override": false}, "their": {"name": "their", "group": "A person", "definition": "person[\"pronouns\"][\"their\"]", "description": "", "templateType": "anything", "can_override": false}, "theirs": {"name": "theirs", "group": "A person", "definition": "person[\"pronouns\"][\"theirs\"]", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "A person", "definition": "if(person[\"gender\"]=\"neutral\",\"\",\"s\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["payment", "weeks"], "variable_groups": [{"name": "A person", "variables": ["person", "pname", "they", "their", "theirs", "s"]}], "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": "How much does {pname} get paid per week?
\n£[[0]]/week
", "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": "{payment}/{weeks}", "maxValue": "{payment}/{weeks}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Cumulative percent decrease", "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/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "tags": ["decrease", "percentages", "taxonomy"], "variablesTest": {"condition": "", "maxRuns": "1000"}, "variables": {"test": {"definition": "precround(precround(price*((100-perc)/100)^5, 2)*((100-perc)/100)^(n-5), 2)", "description": "Calculated value of price2 to ensure we mention rounding errors in advice only when needed.
", "templateType": "anything", "name": "test", "group": "Part b)"}, "pricee3": {"definition": "precround(price*((100 - perc)/100)^(testn-2),2)", "description": "", "templateType": "anything", "name": "pricee3", "group": "Part b)"}, "person": {"definition": "random_person()", "description": "", "templateType": "anything", "name": "person", "group": "Part b)"}, "testn": {"definition": "random(6..9)", "description": "Number of months in total.
", "templateType": "anything", "name": "testn", "group": "Part b)"}, "pricee1": {"definition": "precround(price*((100 - perc)/100)^(testn),2)", "description": "The resulting price after the total of testn months.
", "templateType": "anything", "name": "pricee1", "group": "Part b)"}, "n": {"definition": "if(pricee2 < threshold, testn-1, testn)", "description": "", "templateType": "anything", "name": "n", "group": "Part b)"}, "threshold": {"definition": "siground(pricee1+5,2)", "description": "", "templateType": "anything", "name": "threshold", "group": "Part b)"}, "price": {"definition": "random(300..800) + 0.99", "description": "The original price.
", "templateType": "anything", "name": "price", "group": "Part a)"}, "price2": {"definition": "if(pricee2 < threshold, pricee2, pricee1)", "description": "", "templateType": "anything", "name": "price2", "group": "Part b)"}, "perc": {"definition": "random(2..4 #0.5)", "description": "Percentage decrease per month.
", "templateType": "anything", "name": "perc", "group": "Part a)"}, "pricee2": {"definition": "precround(price*((100 - perc)/100)^(testn-1),2)", "description": "", "templateType": "anything", "name": "pricee2", "group": "Part b)"}}, "statement": "A smartphone's value decreases by $\\var{perc}$% every month. The original price when it is released is $£\\var{price}$.
", "variable_groups": [{"name": "Part a)", "variables": ["price", "perc"]}, {"name": "Part b)", "variables": ["threshold", "pricee1", "pricee2", "pricee3", "testn", "test", "price2", "n", "person"]}], "parts": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "gaps": [{"correctAnswerFraction": false, "precisionMessage": "Round your answer to $2$ decimal places.
", "precisionPartialCredit": 0, "scripts": {}, "maxValue": "precround(price*((100-perc)/100)^5, 2)", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precision": "2", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "precround(price*((100-perc)/100)^5, 2)", "showPrecisionHint": true, "marks": "2", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "type": "numberentry"}], "showFeedbackIcon": true, "prompt": "How much will the smartphone be worth after $5$ months?
\n£ [[0]]
", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "The original price of the phone is $£\\var{price}$ and we are told that the price decreases by $\\var{perc}$% every month.
\n", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "maxValue": "1-{perc}/100", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "prompt": "What is the decimal multiplier for the decrease in the smartphones each month?
", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "1-{perc}/100", "variableReplacements": [], "marks": "0.5", "showCorrectAnswer": true, "type": "numberentry"}, {"precisionMessage": "Round your answer to $2$ decimal places.
", "precisionPartialCredit": 0, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "Multiply the original price by the decimal multiplier to obtain the price after 1 month.
", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "{price}*(1-{perc}/100)", "variableReplacements": [], "marks": "0.5", "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "precision": "2", "maxValue": "{price}*(1-{perc}/100)", "precisionType": "dp", "correctAnswerStyle": "plain", "showPrecisionHint": true}, {"precisionMessage": "Round your answer to $2$ decimal places.
", "precisionPartialCredit": 0, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "Multiply your answer above by the decimal multiplier to obtain the price after 2 months.
\nNote that this is the same as multiplying the original price by $d^2$, where $d$ is the decimal multiplier.
", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "{price}*(1-{perc}/100)^2", "variableReplacements": [], "marks": "0.5", "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "precision": "2", "maxValue": "{price}*(1-{perc}/100)^2", "precisionType": "dp", "correctAnswerStyle": "plain", "showPrecisionHint": true}], "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "maxValue": "n-5", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "n-5", "variableReplacements": [], "marks": "2", "showCorrectAnswer": true, "type": "numberentry"}], "showFeedbackIcon": true, "prompt": "{person['name']} has $£\\var{threshold}$ to spend on a smartphone. After how many more full months will {person['pronouns']['they']} be able to afford the smartphone?
\n[[0]] months
", "marks": 0}], "ungrouped_variables": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given the original price of a smartphone and the rate at which it decreases, calculate its price after a given number of months. In the second part, calculate the time remaining until the price goes below a certain point.
"}, "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "We can use the multiplier method to calculate the new price. If the price decreases by {perc}%, this its value is {100-perc}% of the original value after 1 month. The decimal multiplier for {100-perc}% is
\n\\[\\frac{\\var{100-perc}}{100} = \\var{(100-perc)/100} \\text{.}\\]
\nEach month our smartphone's value can be found by multiplying the previous month's value by the decimal multiplier. For example, after the first month, the value is
\n\\[ \\var{(100-perc)/100} \\times\\mathrm{£}\\var{price} = \\mathrm{£}\\var{dpformat(price*(100-perc)/100,2)}\\text{.} \\]
\nTo calculate the price after 5 months, we multiply the original price of the smartphone by our multiplier 5 times:
\n\\[ \\begin{align} \\text{Final worth} &= \\var{price} \\times \\var{(100-perc)/100} \\times \\var{(100-perc)/100} \\times \\var{(100-perc)/100} \\times \\var{(100-perc)/100} \\times \\var{(100-perc)/100} \\\\&= \\var{price} \\times \\var{(100-perc)/100}^{5} \\\\&= £\\var{precround(price*((100-perc)/100)^5, 2)} {.} \\end{align}\\]
\nFrom part a), the value after 5 months is £$\\var{precround(price*((100-perc)/100)^5, 2)}$. Continuing to multiply the price by the decimal multiplier,
\n\\[£\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100} = £\\var{precround(precround(price*((100-perc)/100)^5, 2)*(100-perc)/100, 2)}\\]
\n\\[£\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^2 = £\\var{precround(precround(price*((100-perc)/100)^6, 2)*(100-perc)/100, 2)}\\]
\n\\[£\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^3 = £\\var{precround(precround(price*((100-perc)/100)^7, 2)*(100-perc)/100, 2)}\\]
\n\\[£\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^4 = £\\var{precround(precround(price*((100-perc)/100)^8, 2)*(100-perc)/100, 2)}\\]
\n\nThe smartphone's value will be below $£\\var{threshold}$ after {n-5} more months ({n} months in total since its release).
"}, {"name": "Simple interest", "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/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "metadata": {"description": "Calculate the interest accrued in a savings account, given the initial balance and annual interest rate.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Suppose you put £{money} into a savings account exactly {years} years ago and you haven't touched the money since. The simple interest rate on the account is {perc2}% per year.
", "variables": {"perc2": {"name": "perc2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(0..2.5 #0.05)"}, "money": {"name": "money", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(100..5000 #100)"}, "years": {"name": "years", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..6)"}}, "tags": ["interests", "percentages", "taxonomy"], "ungrouped_variables": ["perc2", "money", "years"], "functions": {}, "preamble": {"js": "", "css": ""}, "advice": "This is simple interest, which means the amount added each year is a percentage of the original amount. The amount we add is fixed for all {years} years.
\nFirst, we work out the amount of interest for one year:
\n\\begin{align}
\\var{perc2} \\text{% of } \\var{money} &= \\frac{\\var{perc2}}{100} \\times \\var{money} \\\\
&= \\var{perc2/100} \\times \\var{money} \\\\
&= £\\var{dpformat(perc2/100*money,2)} \\text{.}
\\end{align}
The money has been in the account for {years} years, so we multiply $£\\var{dpformat(perc2/100*money,2)}$ by $\\var{years}$.
\n\\[ £\\var{dpformat(perc2/100*money,2)} \\times \\var{years} = £\\var{dpformat(perc2/100*money*years,2)} \\text{.} \\]
\nAdding this to the original balance:
\n\\[ £\\var{money} + £\\var{dpformat(perc2/100*money*years,2)} = £\\var{dpformat(perc2/100*money*years + money,2)} \\text{.} \\]
\nThis is the amount we would get if we withdrew the whole savings balance today.
", "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "precision": "2", "variableReplacements": [], "mustBeReducedPC": 0, "precisionPartialCredit": 0, "minValue": "(1 + perc2/100*years)*money", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "precisionType": "dp", "scripts": {}, "maxValue": "(1 + perc2/100*years)*money", "showCorrectAnswer": false, "strictPrecision": true, "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "marks": "2", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision."}], "type": "gapfill", "marks": 0, "prompt": "If you were to withdraw the money from this account now, how much would you have?
\n£ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}]}]}, {"name": "Trigonometry", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle C calculated from A,B\n pi-aa0-bb0\n ", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(13..29)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a3^2+b3^2))+1", "description": "", "name": "c31"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "check": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA0-BB0-CC0", "description": "", "name": "check"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c31..c32)", "description": "", "name": "c3"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": "", "name": "q3"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle A\n precround(arccos(p0),4)\n ", "description": "", "name": "aa0"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle A calculated from B,C\n pi-bb0-cc0\n ", "description": "", "name": "aa1"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle C\n precround(arccos(r0),4)\n ", "description": "", "name": "cc0"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle B calculated from A,C\n pi-aa0-cc0\n ", "description": "", "name": "bb1"}, "c01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a0^2+b0^2))+1", "description": "", "name": "c01"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "cc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r3),4)", "description": "", "name": "cc3"}, "bb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q3),4)", "description": "", "name": "bb3"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": "", "name": "p3"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA3)", "description": "", "name": "s3"}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a0+0.9*b0,b0+0.9*a0))", "description": "", "name": "c02"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle B\n precround(arccos(q0),4)\n ", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c01..c02)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a0"}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa0)", "description": "", "name": "s0"}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(cc1,3)", "description": "", "name": "cc2"}, "b0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(14..30)", "description": "", "name": "b0"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb2)", "description": "", "name": "t2"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a3"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a3+0.9*b3,b3+0.9*a3))", "description": "", "name": "c32"}}, "ungrouped_variables": ["s3", "cc0", "b0", "cc3", "b3", "cc2", "check", "q0", "q3", "cc5", "s2", "s0", "cc1", "u0", "u3", "u2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "c31", "c32", "a0", "a3", "s5", "c3", "c0", "c02", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$a=\\var{a0}$, $b=\\var{b0}$, $C=\\var{CC0}$
\nSide length $c=$ [[0]]
", "steps": [{"type": "information", "prompt": "Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$. Take care over the fact that $\\cos(\\var{cc0})$ is negative.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\nSuppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\nGiven the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n![](\"resources/images/Triangle_2.PNG\"/)
I want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$.
\n\\[c^2=\\var{a0}^2+\\var{b0}^2-2 \\times \\var{a0}\\times\\var{b0} \\times \\cos (\\var{cc0})=\\var{a0^2}+\\var{b0^2}-\\var{2*a0*b0} \\times (\\var{cos (cc0)})\\]
\n\\[=\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}.\\]
\nHence $a=\\sqrt{\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}}=\\var{sqrt(a0^2+b0^2-2*a0*b0* cos (cc0))}$. To the nearest integer, this is $\\var{c0}$.
"}, {"name": "Apply the sine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-bb0", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..20)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x4))", "description": "", "name": "c31"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(sqrt(x2))", "description": "", "name": "c2"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c4..c5 except 0)", "description": "", "name": "c3"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": "", "name": "q3"}, "x4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(a3^2-b3^2)", "description": "", "name": "x4"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p0),4)", "description": "", "name": "aa0"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-cc0", "description": "", "name": "bb1"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-bb0-cc0", "description": "", "name": "aa1"}, "x5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3^2+b3^2", "description": "", "name": "x5"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r0),4)", "description": "", "name": "cc0"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(c01,c02)", "description": "", "name": "c1"}, "bb5": {"templateType": "anything", "group": 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\nSide length $b=$ [[0]]
\nAngle $C=$ [[1]]
\nSide length $c=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 1, "showPrecisionHint": false}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{AA5}-0.001", "maxValue": "{AA5}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{a3}", "maxValue": "{a3}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n$B=\\var{BB3}$, $C=\\var{CC3}$, $c=\\var{c3}$
\nSide length $b=$ [[0]]
\nAngle $A=$ [[1]]
\nSide length $a=$ [[2]]
\n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nSuppose that $\\Delta ABC$ is a triangle with all interior angles $< \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\nGiven the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n![](\"resources/images/Triangle_1.PNG\"/)
I want acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|<c^2<a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2<c^2<a^2+b^2$ and $b^2-a^2<c^2<a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} > 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.
\nThe second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Two questions testing the application of the Sine Rule when given two angles and a side. In this question, the triangle is always acute.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t0}}{\\var{s0}}=\\var{a0*t0/s0}$. The closest integer is then $\\var{b0}$.
\nSince $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC1}$. To 3dp, this gives $\\var{CC2}$.
\nWe use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u2}}{\\var{s0}}=\\var{a0*u2/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC1}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.
\nb) We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t3}}{\\var{u3}}=\\var{c3*t3/u3}$. The closest integer is then $\\var{b3}$.
\nSince $A+B+C=\\pi$, we calculate $A=\\pi-B-C=\\var{AA4}$. To 3dp, this gives $\\var{AA5}$.
\nWe use the Sine Rule to find $a$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $a=\\dfrac{c \\sin A}{\\sin C}=\\dfrac{\\var{c3}* \\var{s5}}{\\var{u3}}=\\var{c3*s5/u3}$. The closest integer is then $\\var{a3}$. Note that this solution uses the 3dp value of $A$; the answer using $\\var{AA4}$ would give a slightly different long decimal value of $a$, but the integer value would be the same.
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