Skills Audit for Maths and Stats for MAS003 SEFY students without a prerequisite of A-level Maths.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []], "questions": [{"name": "NA11 - Convert Units - Volume - ml to cubic cm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "\"Convert\" from millilitres to cubic centimeters.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Express {x} millilitres ($ml$) in cubic centimetres ($cm^3$).
", "advice": "$1 ml$ is the same measurement of volume as $1 cm^3$ so there is nothing to do to convert except change the units.
\n\nUse this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Given that $1$ $inch$ is approximately $2.54$ $cm$. How long is a $\\var{x}$ inch ruler in $cm$?
", "advice": "The information given in the question tells you to use a conversion factor of $2.54$. So you must multiply $\\var{x}$ by $2.54$ to find the length of the ruler in $cm$.
\n\nUse this link to find some resources which will help you revise this topic.
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\nGive your answer to 2 decimal places.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "The symbol is a comparitor that $<$ says that the object to its left is less than the object to its right. Similarly, $>$ says the object to its left is greater than the object to its right.
\nIn each case the sentence can be read in two ways e.g. $a<b$ can be read as \"$a$ is less than $b$\" OR \"$b$ is greater than $a$\".
\nThe symbol $\\leq$ just changes the sentence to include \"...or equal to...\". This can be of particular relevance when dealing with integers (whole numbers) e.g. $x \\geq 4$ and $x$ is a whole number means that $x$ could be $4, 5, 6,$ or $7$ and so on. Whereas, $x>4$ and $x$ is a whole number means that $x$ could be $5,6,$... etc. Notably in the second case $x$ cannot be $4$.
\n\n
Use this link to find some resources which will help you revise this topic.
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\n\\(\\var{disp_values2[0]}\\var{disp_op2}\\var{disp_values2[1]}\\)
\n", "settings": {"correct_answer_expr": "eval(expression(disp_values2[0]+if(disp_op2=\"\\\\lt\",\"<\",\"<=\")+disp_values2[1]))"}}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NB4 - HCF", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Calculating the LCM and HCF of numbers by using prime factorisation.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "By considering the prime factorisation of $\\var{x}$ and $\\var{y}$, or otherwise, find the highest common factor (HCF) of $\\var{x}$ and $\\var{y}$.
", "advice": "We can write $\\var{x}$ and $\\var{y}$ as a product of prime factors as follows:
\n$\\var{x}=\\var{show_factors(x)}$
\n$\\var{y}=\\var{show_factors(y)}$
\n\nFor HCF of $\\var{x}$ and $\\var{y}$ we need to multiply each prime factor the least number of times it occurs in either $\\var{x}$ or $\\var{y}$
\ni.e. HCF$(x,y) = \\var{show_factors(hcf_xy)}=\\var{hcf_xy}$
\n\nUse this link to find some resources which will help you revise this topic.
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", "advice": "We can write $\\var{x}$ and $\\var{y}$ as a product of prime factors as follows:
\n$\\var{x}=\\var{show_factors(x)}$
\n$\\var{y}=\\var{show_factors(y)}$.
\n\nFor LCM of $\\var{x}$ and $\\var{y}$ we need to multiply each factor the greatest number of times it occurs in either $\\var{x}$ or $\\var{y}$.
\ni.e. LCM$(x,y) = \\var{show_factors(lcm_xy)}=\\var{lcm_xy}$.
\n\nUse this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following expression:
", "advice": "BIDMAS stands for:
\nBrackets
\nIndices
\nDivision
\nMultiplication
\nAddition
\nSubtraction
\n\nAnd is a way for us to remember guidance about the order in which calculations are carried out to ensure that everyone doing teh same sum gets the same answer. In this case the first thing that is in the question is Division.
\nFirst work through the expression from left to right, evaluating any division as you come to it. You should be left with an expression involving only pluses and minuses. Evaluate this expression, again working from left to right. Thus:
\n\n\\[\\var{h}-\\var{a2*b2} \\div \\var{b2}\\]
\n\\[=\\var{h}-\\var{a2}\\]
\n\\[=\\var{h-a2}\\]
\nUse this link to find some resources which will help you revise this topic.
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", "minValue": "{h-a2}", "maxValue": "{h-a2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "ND1 Rounding DP", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": [], "metadata": {"description": "Round numbers to a given number of decimal places.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "We can approximate numbers by rounding.
\nRound $\\var{c1}$ to a given number of decimal places.
", "advice": "The first thing to do when we are rounding numbers is to identify the last digit we are keeping.
\nWhen you're asked to round your answer to a number of decimal places, you need to decide whether to keep the last digit the same (rounding down) or increase it by 1 (rounding up). If the following digit is less than 5 we round down and we round up when the next digit is 5 or more.
\nTo write it down in steps:
\nIt is important to keep in mind that if the digit we are increasing is 9, it becomes zero and we increase the previous digit instead. If this digit is 9 as well, we move along to the left side until we find a digit less than 9.
\nTo round a number to a given number $n$ of decimal places, we look at the $n$th digit after the decimal point.
\nWe have $\\var{c1}$.
\ni)
\nWe look at the first digit after the decimal point. This is $\\var{cdig[4]}$ and the following digit is $\\var{cdig[3]}$ so we round updown to get $\\var{precround(c1, 1)}$.
\nii)
\nThe second digit after the decimal point is $\\var{cdig[3]}$. It is followed by $\\var{cdig[2]}$ so we round updown to get $\\var{precround(c1, 2)}$.
\niii)
\nThe 3rd decimal place is $\\var{cdig[2]}$, followed by $\\var{cdig[1]}$. We get $\\var{precround(c1, 3)}$. The 4th decimal place is $\\var{cdig[1]}$, followed by $\\var{cdig[0]}$. We get $\\var{precround(c1, 4)}$.
\nUse this link to find some resources which will help you revise this topic
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\nii) $\\var{c1}$ rounded to 2 decimal places is: [[1]]
\niii) $\\var{c1}$ rounded to {dp} decimal places is: [[2]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "The first thing to do when we are rounding numbers is to identify the last digit we are keeping.
\nWhen you're asked to round your answer to a number of significant figures, you need to decide whether to keep the last digit same (rounding down) or increase it by 1 (rounding up). If the following digit is less than 5 we round down and we round up when the next digit is 5 or more.
\nTo write it down in steps:
\nIt is important to keep in mind that if the digit we are increasing is 9, it becomes zero and we increase the previous digit instead. If this digit is 9 as well, we move along to the left side until we find a digit less than 9.
\nThe last digit we need to keep will depend on how many zeros there are. We don't consider leading zeros to be significant,
i.e. 0.03 and 0.3 both have 1 significant figure (but 0.30 has two significant figures, since the second zero isn't a 'leading' zero).
i)
\nWe round $\\var{d1}$ to 1 significant figure. The first non-zero digit is $\\var{ddig[5]}$. The following digit is $\\var{ddig[4]}$ so we round updown to get $\\var{dpformat(siground(d1, 1), 0)}$.
\nii)
\nWe round $\\var{d1}$ to {sf} significant figures. The first non-zero digit is $\\var{ddig[5]}$. The second following digit is $\\var{ddig[4]}$, the third following digit is $\\var{ddig[3]}$ and the fourth following digit is $\\var{ddig[2]}$. The digit following the last digit we are keeping is $\\var{ddig[3]}$$\\var{ddig[2]}$$\\var{ddig[1]}$, so we round to get $\\var{sigformat(d1, sf)}$. These are our {sf} significant figures.
\n\nUse this link to find some resources which will help you revise this topic.
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\ni) $\\var{d1}$ rounded to 1 significant figure is: [[0]]
\nii) $\\var{d1}$ rounded to {sf} significant figures is: [[1]]
", "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": "precround(siground(d1, 1),0)", "maxValue": "precround(siground(d1, 1),0)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "siground(d1, sf)", "maxValue": "siground(d1, sf)", "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", "type": "question"}, {"name": "ND3 Rounding SF (decimal)", "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/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": ["rounding"], "metadata": {"description": "Round numbers to a given number of significant figures.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "The first thing to do when we are rounding numbers is to identify the last digit we are keeping.
\nWhen you're asked to round your answer to a number of significant figures, you need to decide whether to keep the last digit same (rounding down) or increase it by 1 (rounding up). If the following digit is less than 5 we round down and we round up when the next digit is 5 or more.
\nTo write it down in steps:
\nIt is important to keep in mind that if the digit we are increasing is 9, it becomes zero and we increase the previous digit instead. If this digit is 9 as well, we move along to the left side until we find a digit less than 9.
\nThe last digit we need to keep will depend on how many zeros there are. We don't consider leading zeros to be significant,
i.e. 0.03 and 0.3 both have 1 significant figure (but 0.30 has two significant figures, since the second zero isn't a 'leading' zero).
i)
\nWe round $\\var{e1}$ to 1 significant figure. The first non-zero digit is $\\var{edig[4]}$, followed by $\\var{edig[3]}$. This is lower than 5 so we round downmore than 5 so we round up to get $\\var{sigformat(e1,1)}$.
\nii)
\nWe round $\\var{e1}$ to {sf} significant figures. The first non-zero digit is $\\var{edig[4]}$. The second following digit is $\\var{edig[3]}$, the third following digit is $\\var{edig[2]}$ and the fourth following digit is $\\var{edig[1]}$. The digit following the last digit we are keeping is $\\var{edig[2]}$$\\var{edig[1]}$$\\var{edig[0]}$, so we round to get $\\var{sigformat(e1, sf)}$. These are our {sf} significant figures.
\n\nUse this link to find some resources which will help you revise this topic.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"edig": {"name": "edig", "group": "Ungrouped variables", "definition": "repeat(random(1..9), 5)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "n_from_digits(ddig)", "description": "Random integer.
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\niii) $\\var{e1}$ rounded to 1 significant figure is: [[0]]
\niv) $\\var{e1}$ rounded to {sf} significant figures is: [[1]]
", "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": "siground(e1, 1)", "maxValue": "siground(e1, 1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "siground(e1, sf)", "maxValue": "siground(e1, sf)", "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", "type": "question"}, {"name": "NE5 - Dividing Negatives", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Calculations with negative numbers.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Calculate $(\\var{x})\\div(\\var{y})$.
", "advice": "When we divide two numbers the rule is,
\nIn this calculation we have
\n\\[(\\var{x})\\div(\\var{y})=\\var{x/y}.\\]
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-10..10)*y", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "{x/y}", "maxValue": "{x/y}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NF3 - Percentage change (decrease then increase)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Compound percentage change.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The value of a car is initially {StartingPrice}. If the value decreases by {dec}%, and then increases by {inc}%, what is the final value?
\nGive your answer correct to two decimal places.
", "advice": "There is a {dec}% decrease in price. This means that price after the decrease will be {100-dec}% of the old price.
\n\\[\\frac{\\var{100-dec}}{100} \\times \\var{StartingPrice} = \\var{(100-dec)/100*StartingPrice}\\]
\nThen there is a {inc}% increase in price. This means the final price will be {100+inc}% of the price after the decrease.
\n\\[\\frac{\\var{100+inc}}{100} \\times \\var{(100-dec)/100*StartingPrice} = £\\var{dpformat((100+inc)/100*(100-dec)/100*StartingPrice,2)}\\]
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dec": {"name": "dec", "group": "Ungrouped variables", "definition": "random(1..50)", "description": "", "templateType": "anything", "can_override": false}, "inc": {"name": "inc", "group": "Ungrouped variables", "definition": "random(1..50)", "description": "", "templateType": "anything", "can_override": false}, "FinalPrice": {"name": "FinalPrice", "group": "Ungrouped variables", "definition": "StartingPrice*(1-dec/100)*(1+inc/100)", "description": "", "templateType": "anything", "can_override": false}, "StartingPrice": {"name": "StartingPrice", "group": "Ungrouped variables", "definition": "random(600..8000 # 10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec", "inc", "FinalPrice", "StartingPrice"], "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": "\n£[[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": "FinalPrice", "maxValue": "FinalPrice", "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": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NF4 Reverse percentages", "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/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": ["decrease", "percentages", "taxonomy"], "metadata": {"description": "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}\\]
\n\nUse this link to find some resources which will help you revise this topic.
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", "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": "1", "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", "type": "question"}, {"name": "NF6 - Percentage of amount 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Calculate one number as percentage of another.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "To find the percentage of a number we can use the formula:
\n\\[ \\text{New value } = \\text{Original value } \\times \\text{Percentage in decimal form} \\]
\nFirstly, to convert a percentage into decimal form we need to divide by $100$:
\n\\[ \\var{p} \\% = \\var{p/100} \\]
\nTherefore,
\n\\[ \\begin{split} \\var{p} \\% \\,\\text{ of } \\var{og} &\\,= \\var{og} \\times \\var{p/100} \\\\ &\\,= \\var{ans} \\end{split} \\]
\n\nUse this link to find resources to help you revise how to work out percentages.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "random(1..99)", "description": "", "templateType": "anything", "can_override": false}, "og": {"name": "og", "group": "Ungrouped variables", "definition": "random(101..999)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "og*p*0.01", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "og", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "Find {p}% of {og}
", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NG4 Subtract fractions", "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": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["adding and subtracting fractions", "adding fractions", "converting between decimals and fractions", "converting integers to fractions", "Fractions", "fractions", "integers", "manipulation of fractions", "subtracting fractions", "taxonomy"], "metadata": {"description": "Manipulate fractions in order to add and subtract them. The difficulty escalates through the inclusion of a whole integer and a decimal, which both need to be converted into a fraction before the addition/subtraction can take place.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following additions and subtractions, giving each fraction in its simplest form. Write the numerator (the top number) as negative if the fraction is negative.
", "advice": "$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2.$
\n\nThe two fractions can be individually multiplied to achieve a common denominator of the lowest common multiple, $\\var{lcm2}.$
\n$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}$ becomes $\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}$ and $\\displaystyle\\frac{\\var{h_coprime}}{\\var{j_coprime}}$ becomes $\\displaystyle\\frac{\\var{hlcm2_j}}{\\var{lcm2}}.$
\nWe can now subtract the second fraction from the first.
\n$\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}-\\frac{\\var{hlcm2_j}}{\\var{lcm2}}=\\frac{\\var{flcmhlcm}}{\\var{lcm2}}.$
\n\nFind out more about this topic using our resource.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"hlcm2_j": {"name": "hlcm2_j", "group": "Part b", "definition": "h_coprime*lcm2_j", "description": "PART B
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Several problems involving the multiplication of fractions, with increasingly difficult examples, including a mixed fraction and a squared fraction. The final part is a word problem.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following multiplication, giving the answer in its simplest form.
", "advice": "\nTo multiply $\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$, address the numerators and denominators separately.
\nMultiply the numerators across both fractions.
\n$\\var{a_coprime}\\times\\var{b_coprime}=\\var{ab}$,
\nand then multiply the denominators across both fractions.
\n$\\var{c_coprime}\\times\\var{d_coprime}=\\var{cd}$.
\nThe values of the multiplied numerators and denominators will be the numerator and denominator of the new fraction: $\\displaystyle\\frac{\\var{ab}}{\\var{cd}}$.
\nThis answer may need simplifying down, and to do this, find the greatest common divisor in both the numerator and denominator and divide by this number.
\nThe greatest common divisor of $\\var{ab}$ and $\\var{cd}$ is $\\var{gcd}$.
\nBy using $\\var{gcd}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{ab}/{cd}}$.
\n\nUse this link to find some resources which will help you revise this topic.
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", "templateType": "anything", "can_override": false}, "h_coprime": {"name": "h_coprime", "group": "Part b", "definition": "h/gcd_gh", "description": "", "templateType": "anything", "can_override": false}, "ee": {"name": "ee", "group": "Part d", "definition": "ddcc/4", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Part a", "definition": "random(3,5,7,11)", "description": "Random number from 1 to 12.
", "templateType": "anything", "can_override": false}, "b_coprime": {"name": "b_coprime", "group": "Part a", "definition": "b/gcd_bd", "description": "", "templateType": "anything", "can_override": false}, "l_coprime2": {"name": "l_coprime2", "group": "Part c", "definition": "l_coprime^2/gcd_lcmc", "description": "", "templateType": "anything", "can_override": false}, "k_coprime": {"name": "k_coprime", "group": "Part b", "definition": "k/gcd_kj", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Part b", "definition": "Random(3,5,7,11,13,17)", "description": "Random number between 1 and 20
", "templateType": "anything", "can_override": false}, "dd": {"name": "dd", "group": "Part d", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "gcd_lcmc": {"name": "gcd_lcmc", "group": "Part c", "definition": "gcd((l_coprime)^2,(m_coprime)^2)", "description": "", "templateType": "anything", "can_override": false}, "m_coprime2": {"name": "m_coprime2", "group": "Part c", "definition": "m_coprime^2/gcd_lcmc", "description": "", "templateType": "anything", "can_override": false}, "gcd_lm": {"name": "gcd_lm", "group": "Part c", "definition": "gcd(l,m)", "description": "", "templateType": "anything", "can_override": false}, "ab": {"name": "ab", "group": "Part a", "definition": "a_coprime*b_coprime", "description": "Variable a times variable b
", "templateType": "anything", "can_override": false}, "gcd_bd": {"name": "gcd_bd", "group": "Part a", "definition": "gcd(b,d)", "description": "", "templateType": "anything", "can_override": false}, "gcd2": {"name": "gcd2", "group": "Part b", "definition": "gcd(num,denom)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part a", "variables": ["a", "b", "c", "d", "a_coprime", "b_coprime", "c_coprime", "d_coprime", "gcd_ac", "gcd_bd", "ab", "cd", "gcd"]}, {"name": "Part b", "variables": ["f", "g", "g_coprime", "h", "h_coprime", "gcd_gh", "k", "k_coprime", "j", "j_coprime", "gcd_kj", "fh", "numif", "num", "denom", "gcda", "gcdb", "gcd2"]}, {"name": "Part d", "variables": ["aa", "bb", "cc", "dd", "ddcc", "ee"]}, {"name": "Part c", "variables": ["l", "m", "gcd_lm", "l_coprime", "m_coprime", "gcd_lcmc", "l_coprime2", "m_coprime2"]}], "functions": {}, "preamble": {"js": "", "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"}, "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": "$\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$ =
Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following sums involving division of fractions. Simplify your answers where possible.
", "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\nUse this link to find some resources which will help you revise this topic
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"f4h4": {"name": "f4h4", "group": "Ungrouped variables", "definition": "f4*h4_coprime", "description": "variable f4 times h4.
\nUsed in part c)
", "templateType": "anything", "can_override": false}, "g4_coprime": {"name": "g4_coprime", "group": "Ungrouped variables", "definition": "g4/gcd(g4,h4)", "description": "PART C
", "templateType": "anything", "can_override": false}, "h4": {"name": "h4", "group": "Ungrouped variables", "definition": "random(5..8 except g4)", "description": "Random number but not the same number as variable g4.
\nUsed in part c.
", "templateType": "anything", "can_override": false}, "h3_coprime": {"name": "h3_coprime", "group": "Ungrouped variables", "definition": "h3/gcd(g3,h3)", "description": "PART C
", "templateType": "anything", "can_override": false}, "f_coprime": {"name": "f_coprime", "group": "part a", "definition": "f/gcd(f,g)", "description": "PART A
", "templateType": "anything", "can_override": false}, "g_coprime": {"name": "g_coprime", "group": "part a", "definition": "g/gcd(f,g)", "description": "PART A
", "templateType": "anything", "can_override": false}, "j1_coprime": {"name": "j1_coprime", "group": "Ungrouped variables", "definition": "j1/gcd(h1,j1)", "description": "PART B
", "templateType": "anything", "can_override": false}, "gcd2": {"name": "gcd2", "group": "Ungrouped variables", "definition": "gcd(f1j1,g1h1)", "description": "greatest common divisor of variables f1j1 and g1h1.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "g1_coprime": {"name": "g1_coprime", "group": "Ungrouped variables", "definition": "g1/gcd(f1,g1)", "description": "PART B
", "templateType": "anything", "can_override": false}, "h1_coprime": {"name": "h1_coprime", "group": "Ungrouped variables", "definition": "h1/gcd(h1,j1)", "description": "PART B
", "templateType": "anything", "can_override": false}, "gcd3": {"name": "gcd3", "group": "Ungrouped variables", "definition": "gcd(num,denom)", "description": "greatest common denominator for part c.
", "templateType": "anything", "can_override": false}, "j1": {"name": "j1", "group": "Ungrouped variables", "definition": "random(h1..11 except h1)", "description": "Random number between 2 and 20 and not the same value as variable h1.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "g1h1": {"name": "g1h1", "group": "Ungrouped variables", "definition": "g1_coprime*h1_coprime", "description": "variable g1 times h1.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "part a", "definition": "random(2..10)", "description": "Random number between 2 and 10.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "f4": {"name": "f4", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "Random number.
\nUsed in part c).
", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "Random number between 2 and 20.
\nUsed in part b)
", "templateType": "anything", "can_override": false}, "g3": {"name": "g3", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "Random number.
\nUsed in part c).
", "templateType": "anything", "can_override": false}, "f3h3": {"name": "f3h3", "group": "Ungrouped variables", "definition": "f3*h3_coprime", "description": "variable f3 times h3.
", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "part a", "definition": "random(2..10)", "description": "Random number from 2 to 10.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "gh": {"name": "gh", "group": "part a", "definition": "g_coprime*h_coprime", "description": "variable g times variable h.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "j_coprime": {"name": "j_coprime", "group": "part a", "definition": "j/gcd(h,j)", "description": "PART A
", "templateType": "anything", "can_override": false}, "denom": {"name": "denom", "group": "Ungrouped variables", "definition": "h3_coprime*(f4h4+g4_coprime)", "description": "Unsimplified denominator of part c.
", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "part a", "definition": "random(h..12 except h)", "description": "Random number between 2 and 10 and not the same value as h.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "f1j1": {"name": "f1j1", "group": "Ungrouped variables", "definition": "f1_coprime*j1_coprime", "description": "variable f1 times j1.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "h4_coprime": {"name": "h4_coprime", "group": "Ungrouped variables", "definition": "h4/gcd(g4,h4)", "description": "PART C
", "templateType": "anything", "can_override": false}, "g1": {"name": "g1", "group": "Ungrouped variables", "definition": "random(f1..11 except f1) ", "description": "Random number between 2 and 30 and not the same value as variable f1.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "fj": {"name": "fj", "group": "part a", "definition": "f_coprime*j_coprime", "description": "variable f times variable j.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Ungrouped variables", "definition": "random(1 .. 3#1)", "description": "Random number between 2 and 6.
\nUsed in part c).
", "templateType": "randrange", "can_override": false}, "f1_coprime": {"name": "f1_coprime", "group": "Ungrouped variables", "definition": "f1/gcd(f1,g1)", "description": "PART B
", "templateType": "anything", "can_override": false}, "h3": {"name": "h3", "group": "Ungrouped variables", "definition": "random(5..8)", "description": "Random number and not the same value as variable g3.
\nUsed in part c).
", "templateType": "anything", "can_override": false}, "gcd1": {"name": "gcd1", "group": "part a", "definition": "gcd(fj,gh)", "description": "greatest common divisor of variable fj and gh.
\nUsed in part a).
", "templateType": "anything", "can_override": false}, "g3_coprime": {"name": "g3_coprime", "group": "Ungrouped variables", "definition": "g3/gcd(g3,h3)", "description": "PART C
", "templateType": "anything", "can_override": false}, "h_coprime": {"name": "h_coprime", "group": "part a", "definition": "h/gcd(h,j)", "description": "PART A
", "templateType": "anything", "can_override": false}, "g4": {"name": "g4", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "Random number.
\nUsed in part c).
", "templateType": "anything", "can_override": false}, "h1": {"name": "h1", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "Random number between 2 and 20.
\nUsed in part b).
", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "h4_coprime*(f3h3+g3_coprime)", "description": "numerator of the improper fraction in part c. Unsimplified.
", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "part a", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "g1", "f1_coprime", "g1_coprime", "h1", "j1", "h1_coprime", "j1_coprime", "f1j1", "g1h1", "gcd2", "f3", "g3", "h3", "g3_coprime", "h3_coprime", "f4", "g4", "h4", "g4_coprime", "h4_coprime", "f3h3", "f4h4", "num", "denom", "gcd3"], "variable_groups": [{"name": "part a", "variables": ["g", "f", "f_coprime", "g_coprime", "h", "j", "h_coprime", "j_coprime", "fj", "gh", "gcd1"]}], "functions": {}, "preamble": {"js": "", "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"}, "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": "$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}}=$
This question tests the student's ability to identify equivalent fractions through spotting a fraction which is not equivalent amongst a list of otherwise equivalent fractions. It also tests the students ability to convert mixed numbers into their equivalent improper fractions. It then does the reverse and tests their ability to convert an improper fraction into an equivalent mixed number.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "A mixed number is a number consisting of an integer and a proper fraction, i.e. a number in the form $ a \\displaystyle \\frac{b}{c}$ where $a$ is an integer and $\\displaystyle\\frac{b}{c}$ is a proper fraction: $b$ is smaller than $c$.
\nAn improper fraction is a fraction where the numerator is larger than the denominator, i.e. a number of the form $\\displaystyle\\frac{d}{e}$ where the numerator, $d$, is greater than the denominator, $e$.
\nTo convert an improper fraction into a mixed number, find out how many times the denominator \\var{h_coprime/gcdb} goes into the numerator \\var{num/gcdb}. You can do this by dividing the numerator by the denominator and taking the whole number part or you can just add the denominator to itself until one more addition would make it bigger. This gives us a whole number part of our mixed fraction of \\var{f}.
\nThe numerator of our mixed fraction is what is left from dividing out the whole number. For this question that is $\\var{num/gcdb}-\\var{f*h_coprime}.
\nFinally the denominator of our mixed fraction is just the denominator of the improper fraction.
\n\\[
\\frac{\\var{num/gcdb}}{\\var{h_coprime/gcdb}} = {\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}}\\text{.}
\\]
Use this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"g": {"name": "g", "group": "Ungrouped variables", "definition": "random(1..h except h)", "description": "Random number between 1 and 15
", "templateType": "anything", "can_override": false}, "g_coprime": {"name": "g_coprime", "group": "Ungrouped variables", "definition": "g/gcd_gh", "description": "PART C
", "templateType": "anything", "can_override": false}, "gcd_gh": {"name": "gcd_gh", "group": "Ungrouped variables", "definition": "gcd(g,h)", "description": "PART C
", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "((h_coprime*f)+g_coprime)", "description": "numerator for the improper fraction c(i)
", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(10 .. 24#1)", "description": "Random number between 1 and 24
", "templateType": "randrange", "can_override": false}, "h_coprime": {"name": "h_coprime", "group": "Ungrouped variables", "definition": "h/gcd_gh", "description": "PART C
", "templateType": "anything", "can_override": false}, "gcdb": {"name": "gcdb", "group": "Ungrouped variables", "definition": "gcd({num},{h_coprime})", "description": "gcd of num and h
", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "Random number between 1 and 5
", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f", "g", "h", "gcd_gh", "g_coprime", "h_coprime", "num", "gcdb"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "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"}, "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": "Write the improper fraction as a mixed number and reduce it down to its simplest form.
\n$\\displaystyle{\\frac{\\var{num/gcdb}}{\\var{h_coprime/gcdb}}} = $ [[2]]
Put fractions in size order.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Which of these two fractions is the largest?
", "advice": "To find which is bigger of $\\frac{\\var{top1}}{\\var{bot1}}$ and $\\frac{\\var{top2}}{\\var{bot2}}$ we need them to have the same denominator. A way to do this is to multiply the top and bottom of $\\frac{\\var{top1}}{\\var{bot1}}$ by $\\var{bot2}$ and multiply the top and bottom of $\\frac{\\var{top2}}{\\var{bot2}}$ by {bot1}. This doesn't change the the value of the fractions as this is just like multiplying by one.
\n\\[\\frac{\\var{top1}}{\\var{bot1}}=\\frac{\\var{top1*bot2}}{\\var{bot1*bot2}},\\quad \\frac{\\var{top2}}{\\var{bot2}}=\\frac{\\var{top2*bot1}}{\\var{bot1*bot2}}\\]
\nNow we can easily see which is bigger by comparing the numerator.
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"top1": {"name": "top1", "group": "Ungrouped variables", "definition": "random(1..(bot1-1))", "description": "", "templateType": "anything", "can_override": false}, "top2": {"name": "top2", "group": "Ungrouped variables", "definition": "random(1..(bot2-1))", "description": "", "templateType": "anything", "can_override": false}, "bot1": {"name": "bot1", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "bot2": {"name": "bot2", "group": "Ungrouped variables", "definition": "random(1..10 except bot1)", "description": "", "templateType": "anything", "can_override": false}, "frac1": {"name": "frac1", "group": "Ungrouped variables", "definition": "top1/bot1", "description": "", "templateType": "anything", "can_override": false}, "frac2": {"name": "frac2", "group": "Ungrouped variables", "definition": "top2/bot2", "description": "", "templateType": "anything", "can_override": false}, "Is1Bigger": {"name": "Is1Bigger", "group": "Ungrouped variables", "definition": "award(1,frac1>frac2)", "description": "", "templateType": "anything", "can_override": false}, "Is2Bigger": {"name": "Is2Bigger", "group": "Ungrouped variables", "definition": "award(1,frac2>frac1)", "description": "", "templateType": "anything", "can_override": false}, "Question": {"name": "Question", "group": "Ungrouped variables", "definition": "[string(top1)+\"/\"+string(bot1),string(top2)+\"/\"+string(bot2)]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(frac1-frac2)>0", "maxRuns": "250"}, "ungrouped_variables": ["top1", "top2", "bot1", "bot2", "frac1", "frac2", "Is1Bigger", "Is2Bigger", "Question"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": "Question", "matrix": "[Is1Bigger,Is2Bigger]"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NH1 FDP convert 1 - Percentage into fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Convert a percentage to a fraction.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Convert $\\var{perc}$% into its equivalent fraction expressed in its simplest form.
", "advice": "Percentages can be converted to fractions by treating them as fractions out of $100$:
\n\\[\\frac{\\var{perc}}{100},\\]
\nand then simplifying. In this case giving:
\n\\[\\frac{\\var{ansn}}{\\var{ansd}}\\]
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"perc": {"name": "perc", "group": "Ungrouped variables", "definition": "random(20 .. 90#10)", "description": "", "templateType": "randrange", "can_override": false}, "ansn": {"name": "ansn", "group": "Ungrouped variables", "definition": "perc/GCD(perc,100)", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Ungrouped variables", "definition": "100/GCD(perc,100)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["perc", "ansn", "ansd"], "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": "[[0]] numerator
\n[[1]] denominator
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Numerator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansn", "maxValue": "ansn", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Denominator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansd", "maxValue": "ansd", "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", "type": "question"}, {"name": "NH2 FDP convert 2 - Decimal into fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Convert a decimal to a fraction.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Express $\\var{dec}$ as a fraction in simplest form.
", "advice": "{advice}
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dec": {"name": "dec", "group": "Ungrouped variables", "definition": "random(0.1 .. 0.9#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "ansn": {"name": "ansn", "group": "Ungrouped variables", "definition": "10*dec/GCD(10*dec,10)", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Ungrouped variables", "definition": "10/GCD(10*dec,10)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(GCD(10*dec,10)=1,adviceno,adviceyes)", "description": "", "templateType": "anything", "can_override": false}, "adviceno": {"name": "adviceno", "group": "Ungrouped variables", "definition": "\"Decimals can be converted to fractions using place value. This decimal only has 1 decimal place and therefore finishes in the \\\"tenths\\\" column. Hence, we can write it as:
\\n\\\\[\\\\frac{\\\\var{dec*10}}{10},\\\\]
\\nand then simplifying (if necessary). In this case no simplification is needed.
\"", "description": "", "templateType": "long string", "can_override": false}, "adviceyes": {"name": "adviceyes", "group": "Ungrouped variables", "definition": "\"Decimals can be converted to fractions using place value. This decimal only has 1 decimal place and therefore finishes in the \\\"tenths\\\" column. Hence, we can write it as:
\\n\\\\[\\\\frac{\\\\var{dec*10}}{10},\\\\]
\\nand then simplifying (if necessary). In this case giving:
\\n\\\\[\\\\frac{\\\\var{ansn}}{\\\\var{ansd}}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec", "ansn", "ansd", "advice", "adviceno", "adviceyes"], "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": "[[0]] Numerator
\n--------------
\n[[1]] Denominator
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Numerator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansn", "maxValue": "ansn", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Denominator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansd", "maxValue": "ansd", "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", "type": "question"}, {"name": "NH3 - FDP convert 3 - Fraction into decimal", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Convert a fraction into a decimal.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write $\\frac{\\var{x}}{\\var{y}}$ as a decimal. Round your answer to 3 decimal places.
", "advice": "You can calculate the decimals by hand using long division of $\\var{x}.000$ divided by $\\var{y}$.
\nIn some cases you may be able to simplify the fraction to something that you know the decimal for.
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "random(1..(y-1))", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "x/y", "maxValue": "x/y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "NJ3 - Dividing amounts in ratios", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Dividing amounts in ratios
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The ratio of ethanol to water is {a}:{b} for an experiment. If I have {volWater}ml of water, how much ethanol do I need?
", "advice": "If there is a ratio of {a}:{b} for ethanol:water then that means for every {b}ml of water we need {a}ml of ethanol.
\nIn our experiment there is {volwater}ml of water so to find the amount of ethanol we divide by {b} and then multiply by {a}.
\n\\[\\var{volwater}\\text{ml}\\times\\frac{\\var{a}}{\\var{b}}=\\var{volwater*a/b}\\text{ml}\\]
Use this link to find some resources which will help you revise this topic.
[[0]]ml
", "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": "volwater/b*a", "maxValue": "volwater/b*a", "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", "type": "question"}, {"name": "NK3 - Standard Form (Calculations)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Calculations involving Standard form.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "To divide two numbers in standard form we can calculate the division of each part of the standard form number separately. In general we have,
\n\\[\\frac{x\\times10^j}{y\\times10^k}=\\frac xy\\times\\frac{10^j}{10^k}=\\frac xy\\times 10^{j-k}\\]
\n\nIn this question we therefore have,
\n\\[\\frac{\\var{a}\\times10^{\\var{n}}}{\\var{b}\\times10^{\\var{m}}}=\\frac{\\var{a}}{\\var{b}}\\times\\frac{10^{\\var{n}}}{10^{\\var{m}}}=\\var{aDivBRound}\\times10^\\var{n-m}.\\]
Since {aDivBRound} is less than 1 then our answer isn't in standard form. In this case we need to reduce the exponent by 1 so the final answer is
\n\\[\\var{MantAnsRound}\\times10^{\\var{ExponentAns}}.\\]
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9.9 # 0.1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9.9 # 0.1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "IsADivBLessOne": {"name": "IsADivBLessOne", "group": "Ungrouped variables", "definition": "a/b<1", "description": "", "templateType": "anything", "can_override": false}, "ExponentAns": {"name": "ExponentAns", "group": "Ungrouped variables", "definition": "if(IsADivBLessOne,n-m-1,n-m)", "description": "", "templateType": "anything", "can_override": false}, "MantAns": {"name": "MantAns", "group": "Ungrouped variables", "definition": "if(IsADivBLessOne, a/b*10, a/b)", "description": "", "templateType": "anything", "can_override": false}, "aDivBRound": {"name": "aDivBRound", "group": "Ungrouped variables", "definition": "precround(a/b,2)", "description": "", "templateType": "anything", "can_override": false}, "MantAnsRound": {"name": "MantAnsRound", "group": "Ungrouped variables", "definition": "precround(MantAns,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "m", "IsADivBLessOne", "ExponentAns", "MantAns", "aDivBRound", "MantAnsRound"], "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": "For the equation
\n\\[\\frac{\\var{a}\\times10^{\\var{n}}}{\\var{b}\\times10^{\\var{m}}}=a\\times10^n\\]
\nfind the values of $a$ and $n$ which keep the answer in standard form.
\nGive $a$ to two decimal places.
\n$a=$[[0]]
$n=$[[1]]
perform a calculation involving negative indices.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Evaluate and simplify the following expression:
\n\\[\\frac{\\var{x}^\\var{n}}{\\var{y}^\\var{m}}\\]
", "advice": "To simplify this expression we use the rule $a^{-n}=\\frac1{a^n}$.
\n\\[\\frac{\\var{x}^\\var{n}}{\\var{y}^\\var{m}}=\\frac{\\var{y}^\\var{-m}}{\\var{x}^\\var{-n}}=\\frac{\\var{y^-m}}{\\var{x^-n}}=\\simplify{{y^-m}/{x^-n}}\\]
\n\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-3..-1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-3..-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "x", "y", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "{x^n/y^m}", "maxValue": "{x^n/y^m}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AB2 Collect terms (including squares)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Simple exercise in collecting terms in different powers of \\(x\\)
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "The idea is to collect together and combine any terms that are the same kind of term so:
\n$\\var{b}$ and $\\var{f}$ are ordinary numbers. We can combine them to get $\\var{b+f}$
\nWe can combine $\\var{a}x$ and $\\var{d}x$ to get $\\var{a+d}x$.
\nThere are also $\\var{c}$ times $x^2$. So our answer is:
\n$\\simplify{{c}x^2+{a+d}x+{b+f}}$
\nUse this link to find some resources that will help you revise how to collect like terms.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except -b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\simplify[!collectNumbers]{{a}x+{b}+{c}x^2+{d}x+{f}}$
", "answer": "{c}x^2+({a}+{d})x+({b}+{f})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*x^2+`+-$n`?*x+`+-$n`?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AB3 - Collecting terms (higher powers)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Simple exercise in collecting terms in different powers of \\(x\\)
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Simplify the following expression by combining \"like\" terms.
", "advice": "First we expand the minus sign in the bracket.
\n\\[\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}=\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4+{-f}x+{-e}x^3}\\]
\nThe idea is to collect together and combine any terms that are the same kind of term so:
\n$\\var{b}x$ and $\\var{-f}x$ both have an $x$ term. We can combine them to get $\\var{b-f}x$
\nWe can combine $\\var{a}x^4$ and $\\var{d}x^4$ to get $\\var{a+d}x^4$.
\nWe combine $\\var{c}x^3$ and $\\var{-e}x^3$ to get $\\var{c-e}x^3$. So our answer is:
\n$\\simplify{{a+d}x^4+{c+e}x^3+{b+f}}$
\n\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except b)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except c)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "f", "e"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}$
", "answer": "({a}+{d})x^4+({c}-{e})x^3+({b}-{f})x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*x^4+`+-$n`?*x^3+`+-$n`?*x", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AB4 multiply terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Filling in the blanks from the answer to a simplified expression involving indices.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Find the missing factor in the following statement:
\n\\[ \\var{3*a}x^\\var{b}y^\\var{c} = \\var{a}x^\\var{d}(?)\\]
", "advice": "We can divide the left handside of the expression by the factor given on the right hand side of the expression to work out the missing factor:
\n\\[\\begin{split}
\\var{3*a}x^\\var{b}y^{\\var{c}}&=\\var{a}x^\\var{d}(?)\\\\
\\Rightarrow \\frac{\\var{3*a}x^\\var{b}y^{\\var{c}}}{\\var{a}x^\\var{d}}&=(?)\\\\
\\Rightarrow 3x^{\\var{b}-\\var{d}}y^\\var{c} &=(?),
\\end{split}\\]
which after simplifying gives the answer:
\n\\[(?) = 3x^{\\var{b-d}}y^{\\var{c}}\\]
\nUse this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Expand the expression below by multiplying each of the terms inside the brackets by the term outside. Give the answer in its simplest form.
", "advice": "Expand brackets using the general formula $\\displaystyle a(x+c)=ax+ac$. This means we multiply each term inside the brackets by the term outside the brackets.
\nIt is easy to forget that the sign outside the brackets also needs to be involved in the multiplication so remember that when two of the same sign are multiplied, the resultant term is positive and when opposite signs are multiplied, the result is negative.
\n\\[
\\begin{align}
\\simplify[terms]{{a[7]}x({a[8]}x^2+{a[9]}x)}&=
\\simplify[!collectNumbers]{{a[7]}x{a[8]}x^2+{a[7]}x{a[9]}x}\\\\&=
\\simplify{{a[7]}*{a[8]}x^3+{a[7]}*{a[9]}x^2}\\text{.}
\\end{align}
\\]
Use this link to find resources to help you revise how to expand single brackets
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Factorise $\\var{q2expr}$
\n", "advice": "The two terms share a common factor of $\\var{q2gcd}\\var{latex(q2v[0])}$ which can be factored out.
\nSo $\\var{q2expr} = \\var{q2ans}$
\n\nUse this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Expand the brackets and simplify
", "advice": "To expand the brackets $\\simplify{({a[1]}x+{a[2]})({a[3]}x+{a[4]})}$ We first multiply all the terms in the left bracket by all the terms in the right bracket. This gives us
\n\\[\\var{a[1]}\\times\\var{a[3]}x^2+\\var{a[1]}x\\times\\var{a[4]}+\\var{a[2]}\\times\\var{a[3]}x+\\var{a[2]}\\times\\var{a[4]}=\\var{a[1]*a[3]}x^2+\\var{a[1]*a[4]}x+\\var{a[2]*a[3]}x+\\var{a[2]*a[4]}.\\]
\nWe can then collect the terms to give us the final answer of
\n\\[\\var{a[1]*a[3]}x^2+\\var{a[1]*a[4]+a[2]*a[3]}x+\\var{a[2]*a[4]}.\\]
Use this link to find some resources which will help you revise this topic.
$\\simplify{({a[1]}x+{a[2]})({a[3]}x+{a[4]})}=$[[0]]
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Expand the brackets and simplify
", "advice": "To expand the brackets $\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}$ We first multiply all the terms in the left bracket by all the terms in the right bracket. This gives us
\n\\[\\var{a[1]}x^\\var{b[1]}\\times\\var{a[3]}x^\\var{b[3]}+\\var{a[1]}x^\\var{b[1]}\\times\\var{c[1]}x^\\var{b[4]}+\\var{a[2]}x^\\var{b[2]}\\times\\var{a[3]}x^\\var{b[3]}+\\var{a[2]}x^\\var{b[2]}\\times\\var{c[1]}x^\\var{b[4]}\\]
\nWe can then simplify to give us the final answer of
\n$\\simplify{{a[1]*a[3]}*x^{b[1]+b[3]}+{a[1]*c[1]}*x^{b[1]+b[4]}+{a[2]*a[3]}*x^{b[2]+b[3]}+{a[2]*c[1]}*x^{b[2]+b[4]}}.$
\n
Use this link to find some resources which will help you revise this topic.
$\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}=$[[0]]
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\n", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "What is the highest common factor of $\\var{c[0]}x^\\var{xp[0]}y^\\var{yp[0]}$ and $\\var{c[1]}x^\\var{xp[1]}y^\\var{yp[1]}$?
", "advice": "In order to find the highest common factor of two single term algebraic expressions you can first find the highest common factor of the coefficients.
\n\nIn this case the Highest common factor of $\\var{c[0]}$ and $\\var{c[1]}$ is $\\var{cans}$.
\nThen work through each of the variables (letters) in turn and see what powers of each appear. In the first expression there is $x^\\var{xp[0]}$ and the second expression there is $x^\\var{xp[1]}$. So they both have at least $x^\\var{xpans}$ in them. Similarly, the first expression there is $y^\\var{yp[0]}$ and the second expression there is $y^\\var{yp[1]}$. So they both have at least $y^\\var{ypans}$ in them.
\nHence, the Highest Common Factor (HCF) of the two expressions is:
\n\\[\\var{cans}x^\\var{xpans}y^\\var{ypans}.\\]
\nUse this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "Given $\\simplify{{m}w-{n} = {p}w+{q}}$, we can get all the $w$'s on the left hand side and all the numbers on the right hand side, and then divide both sides by the coefficient of $w$ to get $w$ by itself.
\n\n| \n | \n | \n |
| \n | \n | \n |
| $\\simplify{{m}w+{n}}$ | \n$=$ | \n$\\simplify{{p}w+{q}}$ | \n
| \n | \n | \n |
| $\\simplify[!cancelTerms,unitFactor]{{m}w-{n}-{p}w}$ | \n$=$ | \n$\\simplify[!cancelTerms,unitFactor]{{p}w+{q}-{p}w}$ | \n
| \n | \n | \n |
| $\\simplify{{m-p}w-{n}}$ | \n$=$ | \n$\\var{q}$ | \n
| \n | \n | \n |
| $\\var{m-p}w-\\var{n}+\\var{n}$ | \n$=$ | \n$\\var{q}+\\var{n}$ | \n
| \n | \n | \n |
| $\\var{m-p}w$ | \n$=$ | \n$\\var{q+n}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{m-p}w}{\\var{m-p}}}$ | \n$=$ | \n$\\displaystyle{\\frac{\\var{q+n}}{\\var{m-p}}}$ | \n
| \n | \n | \n |
| $w$ | \n$=$ | \n$\\displaystyle{\\simplify{{q+n}/{m-p}}} = \\var{precround(ansA,1)} \\text{ to 1 dp}$ | \n
Use this link to find resources to help you revise how to solve linear equations
Solve $\\simplify{({m}w-{n}) = {p}w+{q}}$
\n$w=$ [[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": "ansA", "maxValue": "ansA", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": "100", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AC2 Solve Linear equations with fractions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "sean hunte", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3167/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Solve linear equations with unkowns on both sides. Including brackets and fractions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "\nTo solve an equation like
\n$\\displaystyle{\\frac{x+\\var{num1}}{\\var{num2}}+\\frac{x}{\\var{num3}}=\\var{num4}},$
\nthe first thing to deal with is the denominators of the fractions. In order to do that you multiply both sides of the equation by both denominators $\\var{num2}$ and $\\var{num3}$ (or their lowest common multiple to be slightly more efficient). This will give something equivalent to:
\n$\\displaystyle{\\var{num3 + num2} x+\\var{num3*num1} = \\var{num2*num3*num4}.}$
\nThen proceeding by subtracting $\\var{num3*num1} from both sides:
\n$\\displaystyle{\\var{num3 + num2} x = \\var{num2*num3*num4-num3*num1}.}$
\nAnd finally dividing by $\\var{num2+num3}$:
\n$\\displaystyle{x = \\frac{\\var{num2*num3*num4-num3*num1}}{\\var{num2+num3}}.}$
\n
Use this link to find resources to help you revise how to solve linear equations
Solve $\\displaystyle{\\frac{x+\\var{num1}}{\\var{num2}}+\\frac{x}{\\var{num3}}=\\var{num4}}$.
\n$x=$ [[0]]
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Evaluate the following expression,
\n\\[\\simplify{p^{n}+{a}*r*t+{c}},\\]
\nwhen $p = \\var{pval}$, $r = \\var{rval}$, and $t = \\var{tval}$.
", "advice": "In order to evaluate $\\simplify{p^{n}+{a}*r*t+{c}},$ with the given values, $p = \\var{pval}$, $r = \\var{rval}$, and $t = \\var{tval}$, we replace each instance of that letter with its corresponding value and then apply the rules of BIDMAS:
\n\\[\\var{pval}^\\var{n}+\\var{a}\\times \\var{rval} \\times \\var{tval} + \\var{c}\\]
\nWhich gives the answer $\\var{ans}$.
\nFollow this link for more help on tackling these kind of questions.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Solve the simultaneous equations for x and y, giving your answers as integers or fractions, but not decimals.
\n\\[ \\begin{split} \\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &\\,=\\var{c} \\\\ \\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y} &\\,=\\var{c1} \\end{split}\\]
", "advice": "\\[\\begin{split}\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &\\,=\\var{c} \\qquad\\qquad&(1)\\\\ \\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y} &\\,=\\var{c1} \\qquad\\qquad&(2)\\end{split}\\]
\n{advice1}
\n\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..8 except [0,1])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-8..8 except [0,1,a])", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..8 except [0,1])", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..10 except [round(a1*b/a),b,0,1])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-7..7 except 0)", "description": "", "templateType": "anything", "can_override": false}, "aorsb": {"name": "aorsb", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),'subtract','add')", "description": "", "templateType": "anything", "can_override": false}, "torfb": {"name": "torfb", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),'from','to')", "description": "", "templateType": "anything", "can_override": false}, "sgn": {"name": "sgn", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),-1,1)", "description": "", "templateType": "anything", "can_override": false}, "xn": {"name": "xn", "group": "Ungrouped variables", "definition": "c*abs(b1)+sgn*c1*abs(b)", "description": "", "templateType": "anything", "can_override": false}, "xd": {"name": "xd", "group": "Ungrouped variables", "definition": "a*abs(b1)+sgn*a1*abs(b)", "description": "", "templateType": "anything", "can_override": false}, "xsimp": {"name": "xsimp", "group": "Ungrouped variables", "definition": "xn/xd", "description": "", "templateType": "anything", "can_override": false}, "samex": {"name": "samex", "group": "Ungrouped variables", "definition": "\"For these equations, it is easiest to get a solution for $y$ first, due to the $x$-terms having {eqoroppa} coefficients.
\\nIf we {aorsa} equation (2) {torfa} equation (1) this eliminates the $x$-terms leaving us with one equation in terms of $y$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1)})y} &\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgna*(c1)}}\\\\\\\\ \\\\simplify{{b+sgna*(b1)}y} &\\\\,= \\\\simplify{{c+sgna*(c1)}} \\\\\\\\ y &\\\\,= \\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\var{a}x + \\\\var{b} \\\\times \\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} &\\\\,= \\\\var{c} \\\\\\\\ \\\\var{a}x &\\\\,= \\\\simplify[all, !collectNumbers, !noLeadingminus]{{c} - {c*b+b*sgna*(c1)}/{b+sgna*(b1)}} \\\\\\\\ x &\\\\,= \\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "eqoroppb": {"name": "eqoroppb", "group": "Ungrouped variables", "definition": "if(abs(b)*b1=abs(b1)*b,'equal','equal and opposite')", "description": "", "templateType": "anything", "can_override": false}, "eqoroppa": {"name": "eqoroppa", "group": "Ungrouped variables", "definition": "if(abs(a)*a1=abs(a1)*a,'equal','equal and opposite')", "description": "", "templateType": "anything", "can_override": false}, "samey": {"name": "samey", "group": "Ungrouped variables", "definition": "\"For these equations, it is easiest to get a solution for $x$ first, due to the $y$-terms having {eqoroppb} coefficients.
\\nIf we {aorsb} equation (2) {torfb} equation (1) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1)})x} &\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgn*(c1)}}\\\\\\\\ \\\\simplify{{a+sgn*(a1)}x} &\\\\,= \\\\simplify{{c+sgn*(c1)}} \\\\\\\\ x &\\\\,= \\\\simplify[all, fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\var{a} \\\\times\\\\simplify[fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} + \\\\var{b}y &\\\\,= \\\\var{c} \\\\\\\\ \\\\var{b}y &\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{c} - {c*a+a*sgn*(c1)}/{a+sgn*(a1)}} \\\\\\\\ y &\\\\,= \\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "lcmb": {"name": "lcmb", "group": "Ungrouped variables", "definition": "\"To get a solution for $x$, if we multiply equation (2) by $\\\\simplify{{abs(b/b1)}}$ we will have two equations with {eqoroppb} $y$-coefficients:
\\n\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &\\\\,=\\\\var{c} \\\\qquad\\\\qquad&(3)\\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b/b1)}x +{b1*abs(b/b1)}y} &\\\\,=\\\\var{c1*abs(b/b1)} \\\\qquad\\\\qquad&(4)\\\\end{split}\\\\]
\\nIf we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1*abs(b/b1))})x} &\\\\,= \\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgn*(c1*abs(b/b1))}}\\\\\\\\ \\\\simplify{{a+sgn*(a1*abs(b/b1))}x} &\\\\,= \\\\simplify{{c+sgn*(c1*abs(b/b1))}} \\\\\\\\ x &\\\\,= \\\\simplify[all,fractionNumbers]{{c+sgn*(c1*abs(b/b1))}/{a+sgn*(a1*abs(b/b1))}}. \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\var{a}\\\\times\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)}) + {b}y} &\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{b}y} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c)+a*sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)})} \\\\\\\\ \\\\simplify{{b}y} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c+a*sgn*c1*abs(b/b1))/(a+sgn*a1*abs(b/b1))}} \\\\\\\\ y &\\\\,=\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "lcmb1": {"name": "lcmb1", "group": "Ungrouped variables", "definition": "\"To get a solution for $x$, if we multiply equation (1) by $\\\\simplify{{abs(b1/b)}}$ we will have two equations with {eqoroppb} $y$-coefficients:
\\n\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1/b)}x +{b*abs(b1/b)}y} &\\\\,=\\\\var{c*abs(b1/b)} \\\\qquad\\\\qquad&(3) \\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} &\\\\,=\\\\var{c1} \\\\qquad\\\\qquad&(4)\\\\\\\\ \\\\end{split} \\\\]
\\nIf we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({(a*abs(b1/b))}+{sgn*a1})x} &\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(b1/b))}+{sgn*c1}}\\\\\\\\ \\\\simplify{{(a*abs(b1/b))+sgn*a1}x} &\\\\,= \\\\simplify{{(c*abs(b1/b))+sgn*c1}} \\\\\\\\ x &\\\\,= \\\\simplify[all, fractionNumbers]{{(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}}. \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\var{a}\\\\times\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}) + {b}y} &\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{b}y} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c*abs(b1/b))+a*sgn*c1}/{(a*abs(b1/b))+sgn*a1})} \\\\\\\\ \\\\simplify{{b}y} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c*abs(b1/b)+a*sgn*c1)/(a*abs(b1/b)+sgn*a1)}} \\\\\\\\ y &\\\\,=\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "full": {"name": "full", "group": "Ungrouped variables", "definition": "\"To get a solution for $x$, if we multiply equation (1) by $\\\\var{abs(b1)}$ and equation (2) by $\\\\var{abs(b)}$, we will have two equations with {eqoroppb} $y$-coefficients:
\\n\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1)}x+{b*abs(b1)}y} &\\\\,=\\\\var{c*abs(b1)} \\\\qquad\\\\qquad&(3)\\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b)}x +{b1*abs(b)}y} &\\\\,=\\\\var{c1*abs(b)} \\\\qquad\\\\qquad&(4) \\\\end{split}\\\\]
\\nNow, {aorsb} equation (4) {torfb} equation (3) to eliminate the $y$ terms:
\\n\\\\[ \\\\begin{split} (\\\\simplify[!collectNumbers]{{a*abs(b1)} +{sgn*a1*abs(b)}}) x &\\\\,= \\\\simplify[!collectNumbers]{{c*abs(b1)}+{sgn*c1*abs(b)}} \\\\\\\\ \\\\simplify{{a*abs(b1)+sgn*a1*abs(b)}} x &\\\\,= \\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}} .\\\\end{split} \\\\]
\\nSo the solution for $x$ is \\\\[ x=\\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}/{a*abs(b1)+sgn*a1*abs(b)}}.\\\\]
\\nTo obtain a solution for $y$ we can substitute this value of $x$ into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\simplify[noLeadingminus,fractionNumbers,unitFactor]{{a} {xsimp} + {b}y} &\\\\,=\\\\var{c} \\\\\\\\ \\\\var{b}y &\\\\,= \\\\simplify[!collectNumbers,fractionNumbers]{{c}-{a*xsimp}} \\\\\\\\\\\\var{b}y &\\\\,= \\\\simplify[fractionNumbers]{{c-a*xsimp}} \\\\\\\\y &\\\\,= \\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}} \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "aorsa": {"name": "aorsa", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,'subtract','add')", "description": "", "templateType": "anything", "can_override": false}, "torfa": {"name": "torfa", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,'from','to')", "description": "", "templateType": "anything", "can_override": false}, "sgna": {"name": "sgna", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "lcma": {"name": "lcma", "group": "Ungrouped variables", "definition": "\"To get a solution for $y$, if we multiply equation (2) by $\\\\simplify{{abs(a/a1)}}$ we will have two equations with {eqoroppa} $x$-coefficients:
\\n\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &\\\\,=\\\\var{c} \\\\qquad\\\\qquad&(3)\\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(a/a1)}x +{b1*abs(a/a1)}y} &\\\\,=\\\\var{c1*abs(a/a1)} \\\\qquad\\\\qquad&(4)\\\\end{split}\\\\]
\\nIf we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1*abs(a/a1))})y} &\\\\,= \\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgna*(c1*abs(a/a1))}}\\\\\\\\ \\\\simplify{{b+sgna*(b1*abs(a/a1))}y} &\\\\,= \\\\simplify{{c+sgna*(c1*abs(a/a1))}} \\\\\\\\ y &\\\\,= \\\\simplify[all,fractionNumbers]{{c+sgna*(c1*abs(a/a1))}/{b+sgna*(b1*abs(a/a1))}}. \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\times \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} &\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{a}x} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c)+b*sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} \\\\\\\\ \\\\simplify{{a}x} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c+b*sgna*c1*abs(a/a1))/(b+sgna*b1*abs(a/a1))}} \\\\\\\\ x &\\\\,=\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "lcma1": {"name": "lcma1", "group": "Ungrouped variables", "definition": "\"To get a solution for $y$, if we multiply equation (1) by $\\\\simplify{{abs(a1/a)}}$ we will have two equations with {eqoroppa} $x$-coefficients:
\\n\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a*abs(a1/a)}x +{b*abs(a1/a)}y} &\\\\,=\\\\var{c*abs(a1/a)} \\\\qquad\\\\qquad&(3) \\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} &\\\\,=\\\\var{c1} \\\\qquad\\\\qquad&(4) \\\\end{split}\\\\]
\\nIf we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:
\\n\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({(b*abs(a1/a))}+{sgna*b1})y} &\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(a1/a))}+{sgna*c1}}\\\\\\\\ \\\\simplify{{(b*abs(a1/a))+sgna*b1}y} &\\\\,= \\\\simplify{{(c*abs(a1/a))+sgna*c1}} \\\\\\\\ y &\\\\,= \\\\simplify[all, fractionNumbers]{{(c*abs(a1/a))+sgna*c1}/{(b*abs(a1/a))+sgna*b1}}. \\\\end{split} \\\\]
\\n\\nTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain
\\n\\\\[ \\\\begin{split} \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\times \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c*abs(a1/a)+sgna*c1}/{(b*abs(a1/a))+sgna*b1})} &\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{a}x} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c*abs(a1/a))+b*sgna*c1}/{(b*abs(a1/a))+sgna*b1})} \\\\\\\\ \\\\simplify{{a}x} &\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c*abs(a1/a)+b*sgna*c1)/(b*abs(a1/a)+sgna*b1)}} \\\\\\\\ x &\\\\,=\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "advice1": {"name": "advice1", "group": "Ungrouped variables", "definition": "if(abs(b)=abs(b1), {samey},if(abs(a)=abs(a1),{samex},if(lcm(abs(b),abs(b1))=abs(b),{lcmb},if(lcm(abs(b),abs(b1))=abs(b1),{lcmb1},if(lcm(abs(a),abs(a1))=abs(a),{lcma},if(lcm(abs(a),abs(a1))=abs(a1),{lcma1},{full}))))))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(b-b1)>1 and\nabs(a-a1)>1 and\ngcd(a,c)=1 and\ngcd(a1,c1)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "a1", "b1", "c", "c1", "aorsa", "torfa", "aorsb", "torfb", "sgna", "sgn", "xn", "xd", "xsimp", "eqoroppa", "eqoroppb", "advice1", "samey", "samex", "lcmb", "lcmb1", "lcma", "lcma1", "full"], "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": "$x=$ [[0]]
\n$y=$ [[1]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(c-a*xsimp)/b}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AC6 Rearrange Formulae", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Luigi Pivano", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18182/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Rearrange a specific formula. No randomisation.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Rearrange the following equation, to make $y$ the subject:
\n\\[{cy -b = 3x}\\]
", "advice": "In order to rearrange the equation so that it is in terms of $y$, we must first add $b$ to both sides, and then divide both sides of the equation by $c$:
\n\\begin{split} cy-b &= 3x \\\\ cy &= 3x + b \\\\ y &=\\frac{3x+b}{c} \\end{split}
\n\nUse this link to find some resources which will help you revise this topic.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "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": "$y=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(3x+b)/c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "b", "value": ""}, {"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AC7 Solve Linear equations with fractions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "sean hunte", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3167/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Solve linear equations with unkowns on both sides. Including brackets and fractions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "To solve an equation like
\n$\\displaystyle{\\frac{\\var{a}}{y}=\\frac{\\var{b}}{y+\\var{c}}},$
\nthe first thing to deal with is that the unknown ($y$) that you are trying to find is in the denominator (on the bottom) of the fractions. In order to do that you first times by $y$ on both sides and $(y+\\var{c})$ on both sides leading to
\n\\[\\var{a}(y+\\var{c}) = \\var{b}y.\\]
\nFrom here, multiply out the brackets,
\n\\[\\var{a}y +\\var{a*c} = \\var{b}y.\\]
\nNow collect the $y$-terms on one side and the numbers on the other,
\n\\[\\var{a-b}y=\\var{-a*c}.\\]
\nFinally divide by the coefficient of $y$,
\n\\[y=\\frac{\\var{-a*c}}{\\var{a-b}}.\\]
\n\nUse this link to find resources to help you revise how to solve linear equations
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "(-a*c)/(a-b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a<>b AND GCD(-a*c,a-b)<(a-b)", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c", "ans1"], "variable_groups": [{"name": "e", "variables": []}], "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": "Solve $\\displaystyle{\\frac{\\var{a}}{y}=\\frac{\\var{b}}{y+\\var{c}}}$.
\n$y=$ [[0]] (Give your answer as a fraction)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "fraction", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans1", "maxValue": "ans1", "correctAnswerFraction": true, "allowFractions": true, "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", "type": "question"}, {"name": "AD1 Factorising a Quadratic (a=1)", "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": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["factorisation", "Factorisation", "factorising quadratic equations", "Factorising quadratic equations", "taxonomy"], "metadata": {"description": "Factorise three quadratic equations of the form $x^2+bx+c$.
\nThe first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Factorise the following quadratic equation.
", "advice": "Quadratic equations of the form
\n\\[x^2+bx+c=0\\]
\ncan be factorised to create an equation of the form
\n\\[(x+m)(x+n)=0\\text{.}\\]
\nWhen we expand a factorised quadratic expression we obtain
\n\\[(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\\]
\nTo factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.
\n\nWe need to find two values that add together to make $\\var{v3+v4}$ and multiply together to make $\\var{v3*v4}$.
\n\\[\\begin{align}
\\var{v3} \\times \\var{v4}&=\\var{v3*v4}\\\\
\\var{v3}+\\var{v4}&=\\var{v3+v4}\\\\
\\end{align} \\]
So the factorised form of the equation is
\n\\[\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\\]
\n\nUse this link to find some resources which will help you revise this topic
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\n[[0]] $=0$
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The answer is a comma-separated list of numbers.
\nThe list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.
\nYou can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.
", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n if(settings[\"correctAnswerFractions\"],\n map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n ,\n settings[\"correctAnswer\"]\n ),\n settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n \"Enter a list of numbers separated by{settings['separator']}.\",\n \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n true,\n let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n warn(wrong+\" is not a valid number\");\n fail(wrong+\" is not a valid number.\")\n )\n )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n multiply_credit(0.5,\"Not in order\")\n )\n\nincluded:\nmap(\n let(\n num_student,len(filter(x=y,y,interpreted_answer)),\n num_expected,len(filter(x=y,y,expected_numbers)),\n switch(\n num_student=num_expected,\n true,\n num_studentIs every number in the student's list valid?
", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n true,\n let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n warn(wrong+\" is not a valid number\");\n fail(wrong+\" is not a valid number.\")\n )\n )"}, {"name": "is_sorted", "description": "Are the student's answers in ascending order?
", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n multiply_credit(0.5,\"Not in order\")\n )"}, {"name": "included", "description": "Is each number in the expected answer present in the student's list the correct number of times?
", "definition": "map(\n let(\n num_student,len(filter(x=y,y,interpreted_answer)),\n num_expected,len(filter(x=y,y,expected_numbers)),\n switch(\n num_student=num_expected,\n true,\n num_studentTrue if the student's list doesn't contain any numbers that aren't in the expected answer.
", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n true\n ,\n incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n false\n )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n map(\n if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n ,x\n ,bits\n )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "Should the answer be considered as a set, so the number of times an element occurs doesn't matter?
", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "Numbers included in the student's answer that are not in the expected list.
", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Solving a quadratic equation via factorisation (or otherwise) with the $x^2$-term having a coefficient of 1.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Solve the following quadratic equation by factorisation or otherwise:
\n\\[ \\simplify[unitFactor]{x^2+{b}x+{c}=0} \\]
", "advice": "To solve a quadratic equation of the form \\[ x^2+bx+c=0\\] by factorisation, we want to factorise the equation into the form \\[(x+p)(x+q)=0,\\] where $p+q=b$ and $p \\times q = c$.
\nHence, for the equation \\[\\simplify{x^2+{b}x+{c}=0}, \\]
\nthis can be factorised to \\[\\simplify{(x+{p})(x+{q})=0}.\\] This equation is satisfied when either \\[\\simplify{x+{p}=0} \\quad \\text{or} \\quad \\simplify{x+{q}=0}, \\] which implies the solutions to this quadratic equation are \\[ \\simplify{x={-p}} \\quad \\text{and} \\quad \\simplify{x={-q}} .\\]
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "{p+q}", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "{p*q}", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-10..10 except [0,p])", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "[-p,-q]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(p+q)>0", "maxRuns": 100}, "ungrouped_variables": ["b", "c", "p", "q", "sol"], "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": "$x= $[[0]]
", "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "{sol}", "allowFractions": false, "correctAnswerFractions": false, "isSet": false, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AD3 Completing the square", "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": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["complete the square", "completing the square", "taxonomy"], "metadata": {"description": "Rearrange expressions in the form $ax^2+bx+c$ to $a(x+b)^2+c$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".
\nThis can be useful when it isn't obvious how to fully factorise a quadratic equation.
\nRewrite the following expressions in the form \\[(x+b)^2-c\\]
", "advice": "Completing the square works by noticing that
\n\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]
\nSo when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.
\n\nReplace $x^2+\\var{evens2}x$ with $(x+\\var{evens2/2})^2 - \\var{evens2/2}^2$. Remember to keep the $\\var{evens2-evens1}$ term on the end!
\n\\begin{align}
\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &= \\simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\\\
&= \\simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} }
\\end{align}
Use this link to find some resources which will help you revise this topic.
\n$\\simplify {x^2+ {evens2}x +{evens2-evens1}} =$ [[0]]
It doesn't look like you've completed the square.
"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AD4 Quadratics - factorise (a not 1)", "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": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["coefficient of x^2 greater than 1", "factorisation", "Factorisation", "factorising", "factorising quadratic equations", "Factorising quadratic equations", "factorising quadratic equations with x^2 coefficients greater than 1", "taxonomy"], "metadata": {"description": "Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "As this question involves a number greater than $1$ before the $x^2$ value it has a factorised form $(ax+b)(cx+d)$.
\nTo find $a$ and $c$, we need to consider the factors of $\\var{a*c}$.
\nYou may have to test a a few different options before you find one that works. In this case $a$ and $c$ are $\\var{a}$ and $\\var{c}$.
\nThis means our factorised equation must take the form
\n\\[(\\var{a}x+b)(\\var{c}x+d)=0\\text{.}\\]
\nThis expands to
\n\\[ \\simplify{ {a*c}x^2 + ({a}*d+{c}*b)x + a*b} \\]
\nSo we must find two numbers which add together to make $\\var{a*d+b*c}$, and multiply together to make $\\var{b*d}$.
\nTherefore $b$ and $d$ must satisfy
\n\\begin{align}
b \\times d &=\\var{b*d}\\\\
\\simplify{{a}d+{c}b} &= \\var{a*d+b*c}\\text{.}
\\end{align}
$b = \\var{b}$ and $d = \\var{d}$ satisfy these equations:
\n\\begin{align}
\\var{b} \\times \\var{d} &=\\var{b*d}\\\\
\\simplify[]{ {a}*{d} + {b}*{c} } &= \\var{a*d+b*c}
\\end{align}
So the factorised form of the equation is
\n\\[ \\simplify{({a}x+{b})({c}x+{d}) = 0} \\text{.}\\]
\n$\\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\\var{a}x+\\var{b} = 0$ or $\\var{c}x+ \\var{d} = 0$.
\nSo the roots of the equation are $\\var[fractionnumbers]{-b/a}$ and $\\var[fractionnumbers]{-d/c}$.
\n\nUse this link to find some resources which will help you revise this topic.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "last q", "definition": "random(-5..5 except 0)", "description": "$b$ in $(ax+b)(cx+d)$
", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "last q", "definition": "random(2..8 except a)", "description": "$c$ in $(ax+b)(cx+d)$
", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "last q", "definition": "random(2..3)", "description": "$a$ in $(ax+b)(cx+d)$
", "templateType": "anything", "can_override": false}, "roots": {"name": "roots", "group": "last q", "definition": "sort([-b/a,-d/c])", "description": "The roots of the equation
", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "last q", "definition": "random(-8..8 except 0)", "description": "$d$ in $(ax+b)(cx+d)$
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "last q", "variables": ["a", "b", "c", "d", "roots"]}], "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": "Solve the following equation by factorisation to find $x$.
\n$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\\text{.}$
\nInput your answers in ascending order.
\n$x=$ [[0]]
\n$x=$ [[1]]
", "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": "roots[0]", "maxValue": "roots[0]", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "roots[1]", "maxValue": "roots[1]", "correctAnswerFraction": true, "allowFractions": true, "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", "type": "question"}, {"name": "AD5 Difference of two squares", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Factorising a quadratic expression of the form $a^2x^2-b^2$ to $(ax+b)(ax-b)$, using the difference of two squares formula.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Factorise the following quadratic expression:
\n\\[ \\simplify[unitFactor]{{a^2}x^2-{c^2}} \\]
", "advice": "For a quadratic expression of this form we can make use of the Difference of Squares formula, which states that \\[a^2-b^2 = (a+b)(a-b).\\]
\nTherefore,
\n\\[ \\simplify[unitFactor]{{a^2}x^2-{c^2} = ({a}x+{c})({a}x-{c})}. \\]
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..10 except a)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a}x+{c})({a}x-{c})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "($n`?*x+`+-$n)($n`?*x+`+-$n)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AE1 - Algebraic fractions - addition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "Simplify (qx+a)/(rx+b) +/- (sx+c)/(tx+d)
\nx is a randomised variable. a,b,c,d,q,r,s,t are randomised integers. a,b,c,d run from -5 to 5, including 0. q,r,s,t run from -3 to 3, and can be 0 if the constant term is nonzero, but are mostly 1.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Express $\\displaystyle{\\var{te[0]}\\var{sgnl}\\var{te[1]}}$ as a single fraction.
", "advice": "\\[\\begin{align*} \\var{te[0]}\\var{sgnl}\\var{te[1]} &= \\frac{\\var{ndnde[3]}}{\\var{ndnde[3]}}\\times\\var{te[0]}\\var{sgnl}\\frac{\\var{ndnde[1]}}{\\var{ndnde[1]}}\\times\\var{te[1]}\\\\&=\\frac{\\var{cnd[0]}\\var{sgnl}\\var{cnd[1]}}{(\\var{ndnde[1]})(\\var{ndnde[3]})}\\\\&=\\frac{(\\var{cnd[2]})\\var{sgnl}(\\var{cnd[3]})}{(\\var{ndnde[1]})(\\var{ndnde[3]})}\\\\&=\\frac{\\var{cnd[4]}}{(\\var{ndnde[1]})(\\var{ndnde[3]})}\\\\&=\\var{ans} \\end{align*}\\]
\nThere is no benefit in expanding the denominator. In fact, it is best to leave the denominator factorised, because then it is easier to see if the fraction can be simplified.
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"te": {"name": "te", "group": "Ungrouped variables", "definition": "[simplify(expression(\"(\"+ndnd[0]+\")/(\"+ndnd[1]+\")\"),\"all\"),\n simplify(expression(\"(\"+ndnd[2]+\")/(\"+ndnd[3]+\")\"),\"all\")\n]", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "random(\"a\",\"b\",\"c\",\"d\",\"f\",\"g\",\"h\",\"k\",\"m\",\"n\",\"p\",\"q\",\"r\",\"s\",\"t\",\"u\",\"v\",\"w\",\"x\",\"y\",\"z\")", "description": "the variable to use
", "templateType": "anything", "can_override": false}, "xc": {"name": "xc", "group": "Ungrouped variables", "definition": "repeat(weighted_random([ [1,0.7] , [random(1..3),0.3] ])\n ,4)", "description": "the x-coefficients
", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "map(\nswitch(xc[i]=0, random(2..5),\n xc[i]=1, random(-5..5 except 0),\n xc[i]=2, random(-5..5 except [-4,-2,2,4]),\n random(-5..5 except [-xc[i],xc[i]])\n),i,[0,1,2,3])", "description": "the constants. Make sure that the constants are coprime with their x-coefficient, and if their x-coefficient is 0, that they are positive. There is a condition in variable testing to ensure that no fraction = 1.
", "templateType": "anything", "can_override": false}, "ndnd": {"name": "ndnd", "group": "Ungrouped variables", "definition": "[\"+\"+c[0],\n xc[1]+\"*\"+v+\"+\"+c[1],\n \"+\"+c[2],\n xc[3]+\"*\"+v+\"+\"+c[3]\n ]", "description": "numerator 1, denominator 1, numerator 2, denominator 2, as strings.
", "templateType": "anything", "can_override": false}, "sgn": {"name": "sgn", "group": "Ungrouped variables", "definition": "random(\"+\",\"-\")", "description": "", "templateType": "anything", "can_override": false}, "sgnl": {"name": "sgnl", "group": "Ungrouped variables", "definition": "latex(sgn)", "description": "for display purposes
", "templateType": "anything", "can_override": false}, "ndnde": {"name": "ndnde", "group": "Ungrouped variables", "definition": "map(simplify(expression(x),\"all\"),x,ndnd)", "description": "", "templateType": "anything", "can_override": false}, "cnd": {"name": "cnd", "group": "Ungrouped variables", "definition": "[simplify(expression(\"(\"+ndnd[3]+\")*(\"+ndnd[0]+\")\"),\"all\"),\nsimplify(expression(\"(\"+ndnd[1]+\")*(\"+ndnd[2]+\")\"),\"all\"),\nsimplify(expression(\"(\"+ndnd[3]+\")*(\"+ndnd[0]+\")\"),[\"expandBrackets\",\"all\"]),\nsimplify(expression(\"(\"+ndnd[1]+\")*(\"+ndnd[2]+\")\"),[\"expandBrackets\",\"all\"]),\nsimplify(expression(\n string(simplify(\n expression(\"(\"+ndnd[3]+\")*(\"+ndnd[0]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"])\n )+\"+\"+\n string(simplify(\n expression(sgn+\"(\"+ndnd[1]+\")*(\"+ndnd[2]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"])\n )\n),[\"expandBrackets\",\"basic\"]),\n \nsimplify(expression(\n string(simplify(expression(\"(\"+ndnd[3]+\")*(\"+ndnd[0]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"]))+ \"+\" +\n string(simplify(expression(sgn+\"(\"+ndnd[1]+\")*(\"+ndnd[2]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"]))\n ),[\"all\",\"!noLeadingMinus\"])\n ]", "description": "The combined numerator and denominator terms:
\n0) numerator term 1, 1) numerator term 2,
\n2) brackets expanded num t1, 3) brackets expanded num t2
\n4) numerator, no brackets
\n5) numerator simplified
\n", "templateType": "anything", "can_override": false}, "ansnum": {"name": "ansnum", "group": "Ungrouped variables", "definition": "simplify(expression(\n string(simplify(expression(\"(\"+ndnd[3]+\")*(\"+ndnd[0]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"]))+ \"+\" +\n string(simplify(expression(sgn+\"(\"+ndnd[1]+\")*(\"+ndnd[2]+\")\"),\n [\"expandBrackets\",\"all\",\"!noLeadingMinus\"]))\n ),[\"all\",\"!noLeadingMinus\"])", "description": "", "templateType": "anything", "can_override": false}, "ansden": {"name": "ansden", "group": "Ungrouped variables", "definition": "simplify(expression(\"(\"+ndnd[1]+\")*(\"+ndnd[3]+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "expression(\"(\"+string(ansnum)+\")/(\"+string(ansden)+\")\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "(xc[0]<>xc[1] or c[0]<>c[1]) and (xc[2]<>xc[3] or c[2]<>c[3])", "maxRuns": "76"}, "ungrouped_variables": ["v", "xc", "c", "ndnd", "te", "sgn", "sgnl", "ndnde", "cnd", "ansnum", "ansden", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "?`+/?`+", "partialCredit": 0, "message": "You need to give your answer as just one fraction", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AE2 Algebraic Fractions - addition (harder)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "Simplify the sum of two algebraic fractions where spotting factorising of both numerators and denominators can reduce the work massively.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write the following as a single fraction $\\frac{\\var{num1}}{\\var{den1}}+\\frac{\\var{num2}}{\\var{den2}}$ simplifying as much as possible. Your answer should be in the form $\\frac{\\alpha\\var{v}+\\beta}{\\delta\\var{v}^2-\\gamma}.$
", "advice": "To write the following as a single fraction $\\frac{\\var{num1}}{\\var{den1}}+\\frac{\\var{num2}}{\\var{den2}}$ first factorise as much as possible and look for any cancellations:
\n\\[\\begin{split}
&\\frac{\\var{a}\\times\\var{b}}{\\var{den1fact}} + \\frac{\\var{num2}}{\\var{den2fact}}\\\\
& = \\frac{\\var{b}}{\\var{den1simp}} + \\frac{1}{\\var{f1c}}.
\\end{split}\\]
Then get a common denominator for the two fractions and combine into a single fraction:
\n\\[\\begin{split}
&\\frac{\\var{b}}{\\var{den1simp}} + \\frac{\\var{f1}}{\\var{den1simp}}\\\\
& = \\frac{\\var{b}+\\var{f1}}{\\var{den1simp}}\\\\
& = \\var{ans}.
\\end{split}\\]
Use this link to find some resources which will help you revise this topic.
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Cancelling algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}, {"name": "Anna Strzelecka", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2945/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Simplify the following algebraic expression.
", "advice": "\\[\\frac{{\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}}{{\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}}\\]
\nIn this question there is a quadratic expression which needs to be factorised into the products of binomials in both the numerator and denominator.
\n\\[\\frac{({\\simplify{n+{e1}}})({\\simplify{n+{e2}}})}{({\\simplify{n+{e1}}})({\\simplify{n+{e3}}})}\\]
\nThe repeated binomials in the numerator and denominator cancel, leaving:
\n\\[\\frac{({\\simplify{n+{e2}}})}{({\\simplify{n+{e3}}})}\\]
\n\nUse this link to find some resources which will help you revise this topic.
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", "answer": "(n+{e2})/(n+{e3})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["^2", "^"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "n", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "AE4 - Multiplication of algebraic fractions 1", "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": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": ["adding and subtracting fractions", "adding fractions", "converting between decimals and fractions", "converting integers to fractions", "Fractions", "fractions", "integers", "manipulation of fractions", "subtracting fractions", "taxonomy"], "metadata": {"description": "Manipulate fractions in order to add and subtract them. The difficulty escalates through the inclusion of a whole integer and a decimal, which both need to be converted into a fraction before the addition/subtraction can take place.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following addition, giving the fraction in its simplest form.
\n\\[\\frac{\\var{a}}{y^\\var{n2}}\\times\\frac{y^\\var{n1}}{\\var{b}}\\]
", "advice": "To multiply two fractions you just mulitply the numerators and multiply the denominators. This means we have,
\n\\[\\frac{\\var{a}}{y^\\var{n2}}\\times\\frac{y^\\var{n1}}{\\var{b}}=\\frac{\\var{a}\\times{y^\\var{n1}}}{y^\\var{n2}\\times\\var{b}}=\\frac{\\var{a/gcd_ab}\\times{y^\\var{n1-n2}}}{\\var{b/gcd_ab}}\\]
\n\n
Use this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{statement}
\nFind $x$.
", "advice": "Only round your final answer to 1 decimal place.
\n{advice}
\nUse this link to find some resources to help you revise how to use pythagoras' theorem.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"setup": {"name": "setup", "group": "Varying q and advice", "definition": "random(1,2)", "description": "", "templateType": "anything", "can_override": false}, "answerside": {"name": "answerside", "group": "Varying q and advice", "definition": "sh1", "description": "", "templateType": "anything", "can_override": false}, "answerhyp": {"name": "answerhyp", "group": "Varying q and advice", "definition": "hyp", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Varying q and advice", "definition": "if(setup=1,answerside,answerhyp)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Varying q and advice", "definition": "if(setup=1,advice1,advice2)", "description": "", "templateType": "anything", "can_override": false}, "advice2": {"name": "advice2", "group": "Varying q and advice", "definition": "\"Avoid using rounded values in calculations and just round for the final answer.
Pythagoras Theorem states that, in a right angled triangle, with hypotenuse $c$:
\\\\[a^2 + b^2 = c^2\\\\]
\\nLet\\'s call the unknown value $x$, therefore we can write:
\\n$a = \\\\var{sh1}$, $b =\\\\var{sh2}$ and $c = x$
\\nSo
\\\\[\\\\var{sh1}^2 + \\\\var{sh2}^2 = x^2\\\\]
and therefore
\\n\\\\[x^2 = \\\\var{sh1^2} + \\\\var{sh2^2}\\\\]
\\\\[x = \\\\sqrt{\\\\var{sh1^2} + \\\\var{sh2^2}}\\\\]
\\\\[x = \\\\sqrt{\\\\var{sh1^2+sh2^2}}\\\\]
$x = \\\\var{hyp}$ to 1 d.p.
Avoid using rounded values in calculations and just round for the final answer.
Pythagoras Theorem states that, in a right angled triangle, with hypotenuse $c$:
\\\\[a^2 + b^2 = c^2\\\\]
\\nLet\\'s call the unknown value $x$, therefore we can write:
\\n$a = x$, $b =\\\\var{sh2}$ and $c = \\\\var{hyp}$
\\nSo
\\n\\\\[x^2 + \\\\var{sh2}^2 = \\\\var{hyp}^2\\\\]
\\nand therefore
\\n\\\\[x^2 = \\\\var{hyp^2} - \\\\var{sh2^2}\\\\]
\\n\\\\[x = \\\\sqrt{\\\\var{hyp^2-sh2^2}}\\\\]
$x = \\\\var{sh1}$ to 1 d.p.
one of two shortest sides for calculations.
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", "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": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "GA3 Area of a circle", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Finding the area of a circle when given the diameter of the circle.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Find the area of a circle with diameter $\\var{d}$ cm giving your answer to 1 decimal place.
\n{geogebra_applet('https://www.geogebra.org/m/ngcchpcj',[d: d])}
", "advice": "To calculate the area of a circle we want to use the formula \\[ A = \\pi r^2, \\]
\nwhere $r$ is the radius of the circle.
\nSo, if the diameter, d, is $\\var{d}$ cm, then the radius is, $r=\\frac{d}{2}=\\var{{d}/2}$ cm, then
\n\\[ \\begin{split} Area &\\,=\\var{{d}/2}^2 \\times \\pi \\text{ cm}^2 \\\\ &\\,= \\simplify[all, fractionNumbers]{{{{d}^2/4}}pi} \\text{ cm}^2 \\\\ &\\,= \\var{precround({d}^2/4*pi,1)} \\text{ cm}^2. \\end{split} \\]
\nUse this link to find some resources to help you revise how to calculate the area of a circle.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(6,8,10,12,14,16,18,20)", "description": "", "templateType": "anything", "can_override": true}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["d", "t"], "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": "$Area=$ [[0]] $\\text{ cm}^2$
", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "precround({{d/2}}^2*pi,1)", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "answer": "precround({{d/2}}^2*pi,1)", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "GA4 Volume of a triangular prism", "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": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "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": "", "advice": "For 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{0.5*w6*h6} \\times \\var{d6} \\\\
&= \\var{0.5*w6*h6*d6}\\, \\mathrm{m}^2\\,.
\\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 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 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": "", "advice": "For 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}
Use this link to find resources to help you revise how to calculate the volume of a cylinder.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d4": {"name": "d4", "group": "Cuboid ", "definition": "random(2..5#1)", "description": "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 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}
Find the diagonal or one side of a rectangle using Pythagoras' theorem.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "What is the height of the rectangle below (all measurements given in $cm$)? Please give your answer to one decimal place.
\n{geogebra_applet('https://www.geogebra.org/m/jk3n6sxh',[base: base, hyp: hyp])}
", "advice": "You can see that the rectangle contains a right-angled triangle. We also have the lengths of the base and the hypoteneuse of the triangle. This means we can use Pythagoras' theorem to calculate the last remaining side of the triangle which is also the height of the rectangle.
\n\\[ \\begin{split} Height &\\, = \\sqrt{hypoteneuse^2 - base^2} \\\\ &\\, = \\sqrt{\\var{hyp}^2-\\var{base}^2} \\\\ &\\, = \\sqrt{\\var{{hyp}^2}-\\var{{base}^2}} \\\\ &\\, = \\sqrt{\\var{{{hyp}^2}-{{base}^2}}}\\\\ &\\, = \\var{ans}\\\\ &\\, = \\var{ansr} \\text{ to 1 d.p.} \\end{split} \\]
\nUse this link to find resources to help you revise Pythagoras' theorem.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"hyp": {"name": "hyp", "group": "Ungrouped variables", "definition": "random(10 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}, "base": {"name": "base", "group": "Ungrouped variables", "definition": "random(3 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "sqrt(hyp^2-base^2)", "description": "", "templateType": "anything", "can_override": false}, "ansr": {"name": "ansr", "group": "Ungrouped variables", "definition": "precround(ans,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["hyp", "base", "ans", "ansr"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Answer", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "[[0]]$cm$ (give your answer to 1 d.p.)
", "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": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "GB1 Trigonometry - missing side", "extensions": ["eukleides"], "custom_part_types": [], "resources": [["question-resources/Picture1_caMIdF1.png", "/srv/numbas/media/question-resources/Picture1_caMIdF1.png"], ["question-resources/Picture2_6KE4ZpW.png", "/srv/numbas/media/question-resources/Picture2_6KE4ZpW.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "David Wishart", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1461/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Draws a triangle based on 3 side lengths. Randomises asking angle or side.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{max_height(25,diagram)}
", "advice": "Avoid using rounded values in calculations and just round for the final answer.
{advice}
In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$\\\\text{Opposite} = \\\\var{bc}$
$\\\\text{Hyptonuse} = \\\\var{ab}$
We have \\'O\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $\\\\sin$ formula:
\\\\[ \\\\sin(\\\\text{Angle}) = \\\\frac{\\\\text{Opposite}}{\\\\text{Hypotenuse}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\sin(x) = \\\\frac{\\\\var{bc}}{\\\\var{ab}}\\\\]
We need to use the \\'inverse $\\\\sin$\\' button on the calculator (also called $\\\\arcsin$ or notated $\\\\sin^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\sin^{-1}(\\\\var{bc}/\\\\var{ab})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arcsin(bc/(ab)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arcsin(bc/(ab)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "cos_a": {"name": "cos_a", "group": "advice", "definition": "\"In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$\\\\text{Adjacent} = \\\\var{ac}$
$\\\\text{Hyptonuse} = \\\\var{ab}$
We have \\'A\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $\\\\cos$ formula:
\\\\[ \\\\cos(\\\\text{Angle}) = \\\\frac{\\\\text{Adjacent}}{\\\\text{Hypotenuse}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\cos(x) = \\\\frac{\\\\var{ac}}{\\\\var{ab}}\\\\]
We need to use the \\'inverse $\\\\cos$\\' button on the calculator (also called $\\\\arccos$ or notated $\\\\cos^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\cos^{-1}(\\\\var{ac}/\\\\var{ab})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arccos(ac/(ab)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arccos(ac/(ab)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "tan_a": {"name": "tan_a", "group": "advice", "definition": "\"In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$\\\\text{Opposite} = \\\\var{bc}$
$\\\\text{Adjacent} = \\\\var{ac}$
We have \\'O\\' and \\'A\\' in SOHCAHTOA, so we know we need to use the $\\\\tan$ formula:
\\\\[ \\\\tan(\\\\text{Angle}) = \\\\frac{\\\\text{Opposite}}{\\\\text{Adjacent}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\tan(x) = \\\\frac{\\\\var{bc}}{\\\\var{ac}}\\\\]
We need to use the \\'inverse $\\\\tan$\\' button on the calculator (also called $\\\\arctan$ or notated $\\\\tan^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\tan^{-1}(\\\\var{bc}/\\\\var{ac})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arctan(bc/(ac)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arctan(bc/(ac)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "sin_bc": {"name": "sin_bc", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$\\\\text{Opposite} = x$
$\\\\text{Hypotenuse} = \\\\var{ab}$
We have \\'O\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $\\\\sin$ formula:
\\\\[ \\\\sin(\\\\text{Angle}) = \\\\frac{\\\\text{Opposite}}{\\\\text{Hypotenuse}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\sin(\\\\var{angle}) = \\\\frac{x}{\\\\var{ab}}\\\\]
and rearrange to give:
\\\\[ x = \\\\var{ab} \\\\times \\\\sin(\\\\var{angle}) \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(ab*sin(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(ab*sin(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "cos_ac": {"name": "cos_ac", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$\\\\text{Hypotenuse} = \\\\var{ab}$
$\\\\text{Adjacent} = x$
We have \\'A\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $\\\\cos$ formula:
\\\\[ \\\\cos(\\\\text{Angle}) = \\\\frac{\\\\text{Adjacent}}{\\\\text{Hypotenuse}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\cos(\\\\var{angle}) = \\\\frac{x}{\\\\var{ab}}\\\\]
and rearrange to give:
\\\\[ x = \\\\var{ab} \\\\times \\\\cos(\\\\var{angle}) \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(ab*cos(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(ab*cos(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "tan_ac": {"name": "tan_ac", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$\\\\text{Opposite} = \\\\var{bc}$
$\\\\text{Adjacent} = x$
We have \\'O\\' and \\'A\\' in SOHCAHTOA, so we know we need to use the $\\\\tan$ formula:
\\\\[ \\\\tan(\\\\text{Angle}) = \\\\frac{\\\\text{Opposite}}{\\\\text{Adjacent}}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ \\\\tan(\\\\var{angle}) = \\\\frac{\\\\var{bc}}{x}\\\\]
and rearrange to give:
\\\\[ x = \\\\frac{\\\\var{bc}}{\\\\tan(\\\\var{angle})} \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(bc/tan(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(bc/tan(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "precround(180*(arcsin(bc/(ab)))/pi,1) = precround(angle,1)", "maxRuns": "6"}, "ungrouped_variables": [], "variable_groups": [{"name": "Unnamed group", "variables": ["ab", "ac", "bc", "diagram", "angle", "SCT", "AngORside", "answer"]}, {"name": "triangle types", "variables": ["d_t_a_2", "d_t_s_1", "d_s_a_1", "d_c_a_1", "d_c_s_1", "d_s_s_1", "d_c_s_2", "d_t_a_1", "d_t_s_2", "d_s_a_2", "d_s_s_2", "d_c_a_2"]}, {"name": "advice", "variables": ["advice", "tan_a", "sin_a", "cos_a", "sin_bc", "cos_ac", "tan_ac"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "Given a right angled triangle as shown calculate the value of x.
\nAngles are given in degrees (make sure you calculator is in the right mode)
Give your answer correct to 2 decimal place.
Draws a triangle based on 3 side lengths. Randomises asking angle or side.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{max_height(25,diagram)}
", "advice": "Avoid using rounded values in calculations and just round for the final answer.
{advice}
In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$Opposite = \\\\var{bc}$
$Hyptonuse = \\\\var{ab}$
We have \\'O\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $sin$ formula:
\\\\[ sin(Angle) = \\\\frac{Opposite}{Hypotenuse}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ sin(x) = \\\\frac{\\\\var{bc}}{\\\\var{ab}}\\\\]
We need to use the \\'inverse sin\\' button on the calculator (also called $arcsin$ or notated $sin^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = arcsin(\\\\var{bc}/\\\\var{ab})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arcsin(bc/(ab)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arcsin(bc/(ab)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "cos_a": {"name": "cos_a", "group": "advice", "definition": "\"In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$Adjacent = \\\\var{ac}$
$Hyptonuse = \\\\var{ab}$
We have \\'A\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $cos$ formula:
\\\\[ cos(Angle) = \\\\frac{Adjacent}{Hypotenuse}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ cos(x) = \\\\frac{\\\\var{ac}}{\\\\var{ab}}\\\\]
We need to use the \\'inverse cos\\' button on the calculator (also called $arccos$ or notated $cos^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = arccos(\\\\var{ac}/\\\\var{ab})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arccos(ac/(ab)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arccos(ac/(ab)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "tan_a": {"name": "tan_a", "group": "advice", "definition": "\"In this situation $x$ is an angle. We label the known sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we are interested in:
\\n$Opposite = \\\\var{bc}$
$Adjacent = \\\\var{ac}$
We have \\'O\\' and \\'A\\' in SOHCAHTOA, so we know we need to use the $tan$ formula:
\\\\[ tan(Angle) = \\\\frac{Opposite}{Adjacent}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ tan(x) = \\\\frac{\\\\var{bc}}{\\\\var{ac}}\\\\]
We need to use the \\'inverse sin\\' button on the calculator (also called $arctan$ or notated $tan^{-1}$) in order to isolate $x$:
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = arctan(\\\\var{bc}/\\\\var{ac})\\\\]
\\n\\\\[ x = \\\\var{precround(180*(arctan(bc/(ac)))/pi,4)}\\\\]
\\nRound as required:
\\n\\\\[x = \\\\var{precround(180*(arctan(bc/(ac)))/pi,2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "sin_bc": {"name": "sin_bc", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$Opposite = x$
$Hypotenuse = \\\\var{ab}$
We have \\'O\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $sin$ formula:
\\\\[ sin(Angle) = \\\\frac{Opposite}{Hypotenuse}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ sin(\\\\var{angle}) = \\\\frac{x}{\\\\var{ab}}\\\\]
and rearrange to give:
\\\\[ x = \\\\var{ab} \\\\times \\\\sin(\\\\var{angle}) \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(ab*sin(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(ab*sin(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "cos_ac": {"name": "cos_ac", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$Hypotenuse = \\\\var{ab}$
$Adjacent = x$
We have \\'A\\' and \\'H\\' in SOHCAHTOA, so we know we need to use the $cos$ formula:
\\\\[ cos(Angle) = \\\\frac{Adjacent}{Hypotenuse}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ cos(\\\\var{angle}) = \\\\frac{x}{\\\\var{ab}}\\\\]
and rearrange to give:
\\\\[ x = \\\\var{ab} \\\\times \\\\cos(\\\\var{angle}) \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(ab*cos(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(ab*cos(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "tan_ac": {"name": "tan_ac", "group": "advice", "definition": "\"In this situation $x$ is a side. We label the relevant sides of the triangle \\'opposite\\', \\'adjacent\\' and \\'hypothenuse\\' in relation to the angle we know:
\\n$Opposite = \\\\var{bc}$
$Adjacent = x$
We have \\'O\\' and \\'A\\' in SOHCAHTOA, so we know we need to use the $tan$ formula:
\\\\[ tan(Angle) = \\\\frac{Opposite}{Adjacent}\\\\]
\\nNow we subsitute the values we have in this particular question
\\n\\\\[ tan(\\\\var{angle}) = \\\\frac{\\\\var{bc}}{x}\\\\]
and rearrange to give:
\\\\[ x = \\\\frac{\\\\var{bc}}{tan(\\\\var{angle})} \\\\]
Make sure your calculator is set to \\'degree\\' mode, if you get an odd answer you are likely in the wrong mode!
\\\\[ x = \\\\var{precround(bc/tan(pi*angle/180),4)}\\\\]
\\nRound as required:
\\n\\\\[ x = \\\\var{precround(bc/tan(pi*angle/180),2)}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "angle": {"name": "angle", "group": "Unnamed group", "definition": "If(SCT='c',arccos(ac/ab),if(SCT = 's',arcsin(bc/ab),arctan(bc/ac)))", "description": "", "templateType": "anything", "can_override": false}, "gen_ac": {"name": "gen_ac", "group": "Unnamed group", "definition": "random(3 .. 12#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "gen_bc": {"name": "gen_bc", "group": "Unnamed group", "definition": "random(5 .. 15#0.1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "300"}, "ungrouped_variables": [], "variable_groups": [{"name": "Unnamed group", "variables": ["ab", "ac", "bc", "diagram", "SCT", "AngORside", "answer", "angle", "gen_ac", "gen_bc"]}, {"name": "triangle types", "variables": ["d_t_a_2", "d_t_s_1", "d_s_a_1", "d_c_a_1", "d_c_s_1", "d_s_s_1", "d_c_s_2", "d_t_a_1", "d_t_s_2", "d_s_a_2", "d_s_s_2", "d_c_a_2"]}, {"name": "advice", "variables": ["advice", "tan_a", "sin_a", "cos_a", "sin_bc", "cos_ac", "tan_ac"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "Given a right angled triangle as shown calculate the value of x.
\n
Give your answer in degrees (make sure you calculator is in the right mode), correct to 2 decimal place.
Draws a triangle based on 3 side lengths. Randomises asking angle or side.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{diagram}
\nFind x.
", "advice": "{Advice}
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Ruleuse": {"name": "Ruleuse", "group": "Question structure", "definition": "random('s','c','s','c')", "description": "", "templateType": "anything", "can_override": false}, "ANGorSIDE": {"name": "ANGorSIDE", "group": "Question structure", "definition": "random('ang','side')", "description": "", "templateType": "anything", "can_override": false}, "cosSIDEadvice": {"name": "cosSIDEadvice", "group": "Question structure", "definition": "\"First recognise that the diagram is a non-right angled triangle and that there are the lengths of two sides given and the angle specifically between those two sides. Further to this, the instruction is to find the other missing side. These are the conditions for when to use the $\\\\textit{cosine rule}$.
\\nThe formula for a missing side using the cosine rule is:
\\n\\\\[ a^2 = b^2 + c^2 - 2bc \\\\cos(A)\\\\]
\\nThe labels of $a$, $b$ and $c$ can be misleading. The critical thing is that regardless of the letters used in the diagram, the $a$ (side) and $A$ (angle) labels are applied to the angle given and it\\'s opposite side.
\\nIn this case:
\\n\\\\[ a=x, \\\\quad b=\\\\var{a}, \\\\quad c=\\\\var{b}, \\\\text{and} \\\\quad A=\\\\var{Cang},\\\\]
\\nwhere the choice of which way round $b$ and $c$ are assigned doesn\\'t matter.
\\nSo, we now have:
\\n\\\\[x^2 = \\\\var{a}^2 +\\\\var{b}^2-2\\\\times\\\\var{a}\\\\times\\\\var{b}\\\\times\\\\cos{(\\\\var{Cang})},\\\\]
\\nhence,
\\n\\\\[x=\\\\sqrt{\\\\var{a^2 +b^2-2*a*b*(cos(Cang))}}\\\\]
\\n\\\\[x=\\\\var{c}\\\\]
\\n\\\\[x=\\\\var{ans}\\\\text{ to 1 decimal place.}\\\\]
\"", "description": "case 1: missing side in the cosine rule.
", "templateType": "long string", "can_override": false}, "cosANGadvice": {"name": "cosANGadvice", "group": "Question structure", "definition": "\"First recognise that the diagram is a non-right angled triangle and that there are the lengths of all three sides given. Further to this, the instruction is to find the a missing angle. These are the conditions for when to use the $\\\\textit{cosine rule}$ but in its rearranged form to find an angle. You need to identify which side is \\\"$a$\\\" as being the one opposite the angle you are asked to find.
\\nThe formula for a missing angle using the cosine rule is:
\\n\\\\[ A = \\\\arccos\\\\left(\\\\frac{b^2+c^2-a^2}{2bc}\\\\right)\\\\]
\\nThe labels of $a$, $b$ and $c$ can be misleading. The critical thing is that regardless of the letters used in the diagram, the $a$ (side) and $A$ (angle) labels are applied to the side opposite the angle that is asked for and the angle that is asked for.
\\nIn this case:
\\n\\\\[ a=\\\\var{c_round}, \\\\quad b=\\\\var{a}, \\\\quad c=\\\\var{b}, \\\\text{and} \\\\quad A= x,\\\\]
\\nwhere the choice of which way round $b$ and $c$ are assigned doesn\\'t matter.
\\nSo, we now have:
\\n\\\\[x = \\\\arccos\\\\left(\\\\frac{\\\\var{a}^2+\\\\var{b}^2-\\\\var{c_round}^2}{2\\\\times\\\\var{a}\\\\times\\\\var{b}}\\\\right),\\\\]
\\nhence,
\\n\\\\[x=\\\\var{(180/pi)*arccos((a^2 +b^2-c_round^2)/(2*a*b))}\\\\]
\\n\\\\[x=\\\\var{ans}\\\\text{ to 1 decimal place.}\\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "sinSIDEadvice": {"name": "sinSIDEadvice", "group": "Question structure", "definition": "\"First recognise that the diagram is a non-right angled triangle and that a single length is provided, along with two angles, crucially including the angle opposite the given side. Further to this, the instruction is to find the a missing angle. These are the conditions for when to use the $\\\\textit{sine rule}$. The sine rule uses the sides and angles in pairs and uses two pairs for any given calculation
\\nThe formula for finding a side using the sine rule can be written as:
\\n\\\\[ \\\\frac{a}{\\\\sin(A)}=\\\\frac{b}{\\\\sin(B)}\\\\]
\\nThe labels of $a$, $b$ and $c$ can be misleading. The critical thing is that regardless of the letters used in the diagram, the side being asked for is in the above notation $a$.
\\nIn this case:
\\n\\\\[ a=x, \\\\quad b=\\\\var{a}, \\\\quad A=\\\\var{Cang}, \\\\text{and} \\\\quad B= \\\\var{Aang_round}.\\\\]
\\nSo, we now have:
\\n\\\\[\\\\frac{x}{\\\\sin{(\\\\var{Cang})}}=\\\\frac{\\\\var{a}}{\\\\sin{(\\\\var{Aang_round})}},\\\\]
\\nhence,
\\n\\\\[x=\\\\frac{\\\\var{a}}{\\\\sin{(\\\\var{Aang_round})}}\\\\times\\\\sin{(\\\\var{Cang})},\\\\]
\\n\\\\[x=\\\\var{ans}\\\\text{ to 1 decimal place.}\\\\]
\"", "description": "case 3
", "templateType": "long string", "can_override": false}, "sinANGadvice": {"name": "sinANGadvice", "group": "Question structure", "definition": "safe(\"First recognise that the diagram is a non-right angled triangle and that two lengths are provided, along with an angle, crucially including an angle opposite a given side. Further to this, the instruction is to find the a missing side. These are the conditions for when to use the $\\\\textit{sine rule}$. The sine rule uses the sides and angles in pairs and uses two pairs for any given calculation
\\nThe formula for finding an angle using the sine rule can be written as:
\\n\\\\[ \\\\frac{\\\\sin(A)}{a}=\\\\frac{\\\\sin(B)}{b}\\\\]
\\nThe labels of $a$, $b$ and $c$ can be misleading. The critical thing is that regardless of the letters used in the diagram, the angle being asked for is in the above notation $A$.
\\nIn this case:
\\n\\\\[ a=\\\\var{c_round}, \\\\quad b=\\\\var{a}, \\\\quad A= x, \\\\text{and} \\\\quad B= \\\\var{Aang_round}.\\\\]
\\nSo, we now have:
\\n\\\\[\\\\frac{\\\\sin{(x)}}{\\\\var{c_round}}=\\\\frac{\\\\sin{(\\\\var{Aang_round})}}{\\\\var{a}},\\\\]
\\nhence,
\\n\\\\[x=\\\\arcsin\\\\left(\\\\var{c_round}\\\\times\\\\frac{\\\\sin{(\\\\var{Aang_round})}}{\\\\var{a}}\\\\right),\\\\]
\\n\\\\[x=\\\\var{ans}\\\\text{ to 1 decimal place.}\\\\]
\")", "description": "case 4
", "templateType": "long string", "can_override": false}, "advice": {"name": "advice", "group": "Question structure", "definition": "If(Ruleuse='c',IF(ANGorSIDE='ang',cosANGadvice,cosSIDEadvice),IF(ANGorSIDE='ang',sinANGadvice,sinSIDEadvice))", "description": "", "templateType": "anything", "can_override": false}, "cosSIDEdiagram": {"name": "cosSIDEdiagram", "group": "Diagrams", "definition": "geogebra_applet('https://www.geogebra.org/m/czffcqgn',[ac: a,bc: b,Cang: Cang])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Quantities", "definition": "random(5 .. 10#0.1)", "description": "side length a
", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Quantities", "definition": "random(5 .. 10#0.1)", "description": "side length b
", "templateType": "randrange", "can_override": false}, "Cang": {"name": "Cang", "group": "Quantities", "definition": "random(40..140 except 85..95)", "description": "C angle in degrees
", "templateType": "anything", "can_override": false}, "cosANGdiagram": {"name": "cosANGdiagram", "group": "Diagrams", "definition": "geogebra_applet('https://www.geogebra.org/m/rn8p6hk9',[ac: a,bc: b,Cang: Cang])", "description": "", "templateType": "anything", "can_override": false}, "sinSIDEdiagram": {"name": "sinSIDEdiagram", "group": "Diagrams", "definition": "geogebra_applet('https://www.geogebra.org/m/qayf6ejk',[ac: a,bc: b,Cang: Cang])", "description": "", "templateType": "anything", "can_override": false}, "sinANGdiagram": {"name": "sinANGdiagram", "group": "Diagrams", "definition": "geogebra_applet('https://www.geogebra.org/m/ghb43tsn',[ac: a,bc: b,Cang: Cang])", "description": "", "templateType": "anything", "can_override": false}, "diagram": {"name": "diagram", "group": "Diagrams", "definition": "If(Ruleuse='c',IF(ANGorSIDE='ang',cosANGdiagram,cosSIDEdiagram),IF(ANGorSIDE='ang',sinANGdiagram,sinSIDEdiagram))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Quantities", "definition": "sqrt(a^2+b^2-2*a*b*cos(Cang*Pi/180))", "description": "", "templateType": "anything", "can_override": false}, "Aang": {"name": "Aang", "group": "Quantities", "definition": "arcsin(a*sin(Cang*Pi/180)/c)*180/pi", "description": "angle A in degrees
", "templateType": "anything", "can_override": false}, "Bang": {"name": "Bang", "group": "Quantities", "definition": "180-(Aang+Cang)", "description": "", "templateType": "anything", "can_override": false}, "cosSIDEans": {"name": "cosSIDEans", "group": "Quantities", "definition": "c", "description": "", "templateType": "anything", "can_override": false}, "cosANGans": {"name": "cosANGans", "group": "Quantities", "definition": "arccos((a^2+b^2-c_round^2)/(2*a*b))*180/pi", "description": "Calculated answer for c from rounded values - as these will be seen information by student.
", "templateType": "anything", "can_override": false}, "c_round": {"name": "c_round", "group": "Quantities", "definition": "precround(c,1)", "description": "", "templateType": "anything", "can_override": false}, "Aang_round": {"name": "Aang_round", "group": "Quantities", "definition": "precround(Aang,1)", "description": "", "templateType": "anything", "can_override": false}, "Bang_round": {"name": "Bang_round", "group": "Quantities", "definition": "precround(Bang,1)", "description": "", "templateType": "anything", "can_override": false}, "Cang_roundcos": {"name": "Cang_roundcos", "group": "Quantities", "definition": "Precround((180/pi)*arccos((a^2+b^2-c_round^2)/(2*a*b)),1)", "description": "", "templateType": "anything", "can_override": false}, "sinANGans": {"name": "sinANGans", "group": "Quantities", "definition": "If(Cang<90,arcsin(c_round*(sin(Aang_round*pi/180)/a))*180/pi,180 - arcsin(c_round*(sin(Aang_round*pi/180)/a))*180/pi)", "description": "", "templateType": "anything", "can_override": false}, "sinSIDEans": {"name": "sinSIDEans", "group": "Quantities", "definition": "(a/sin(aang_round*pi/180))*sin(cang*pi/180)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Quantities", "definition": "precround(If(Ruleuse='c',IF(ANGorSIDE='ang',cosANGans,cosSIDEans),IF(ANGorSIDE='ang',sinANGans,sinSIDEans)),1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a+b>c and b+c>a and a+c>b", "maxRuns": "200"}, "ungrouped_variables": [], "variable_groups": [{"name": "Question structure", "variables": ["Ruleuse", "ANGorSIDE", "cosSIDEadvice", "cosANGadvice", "sinSIDEadvice", "sinANGadvice", "advice"]}, {"name": "Diagrams", "variables": ["cosSIDEdiagram", "cosANGdiagram", "sinSIDEdiagram", "sinANGdiagram", "diagram"]}, {"name": "Quantities", "variables": ["a", "b", "Cang", "c", "Aang", "Bang", "cosSIDEans", "cosANGans", "sinANGans", "sinSIDEans", "c_round", "Aang_round", "Bang_round", "Cang_roundcos", "ans"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Answer", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$x =$[[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": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": "100", "precisionMessage": "", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "CA1 Straight Line Graphs", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "Calculating gradient and finding intercept from a geogebra graph.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "{app}
Find the gradient of the line.
Firstly draw a right angled 'step' from left to right. This triangle can be anywhere, but it is more helpful for it to have corners on the vertices (whole number points) of the graph and it is easier to calculate with postive numbers.
\n{app_advice}
\nBefore we start to calculate, notice that the line is {uod}, so the gradient will be {pon} and the line is {sos}, so the absolute value of the number will be {mol}.
Now find the coordinates of the places your triangle meets the line
$(x_1,y_1)=(\\var{ax},\\var{ay})$ and $(x_2,y_2)=(\\var{bx},\\var{by})$
\nWe need to compare the 'rise on the y-axis' to the 'run across the x-axis', we can say that:
\n$\\text{gradient} = \\frac{\\text{rise}}{\\text{run}}$
\nThis is equivalent to using the formula:
$ m = \\frac{y_2 - y_1}{x_2 - x_1} $
and substitute the coordinates of the vertices of the triangle:
$\\begin{split} &\\, m = \\frac{\\var{by} - \\var{ay}}{\\var{bx} - \\var{ax}} \\\\
&\\, = \\frac{\\var{by-ay}}{\\var{bx-ax}} \\\\
&\\, = \\var[fractionNumbers]{m} \\\\
\\end{split} $
if(m=abs(m),'positive','negative')
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$\\var[fractionNumbers]{m}$
Multiple choice - select the quadratic graph.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Which of the following is the graph $y=x^2$.
", "advice": "Use this link to find some resources to help you familiarise yourself with these graphs.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["{geogebra_applet('https://www.geogebra.org/m/tpfzv3w7')}", "{geogebra_applet('https://www.geogebra.org/m/zftpwq64')}", "{geogebra_applet('https://www.geogebra.org/m/we3gngqa')}", "{geogebra_applet('https://www.geogebra.org/m/cadkup6r')}"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "SA5 Interpret a Box Plot", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Michael Pan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23528/"}], "tags": [], "metadata": {"description": "Interpreting the elements of a box plot
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The diagram below shows a box plot of some data.
\n{geogebra_applet{\"https://www.geogebra.org/m/aj2hcbhg\",[lv: lv,lq: lq,m: m,uq: uq,hv: hv]}}
\n", "advice": "A boxplot (also known as a box-and-whisker diagram or plot) is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation (sample minimum), lower quartile (Q1), median (Q2), upper quartile (Q3), and largest observation (sample maximum). A boxplot may also indicate which observations, if any, might be considered outliers.
\nFor more information on box plots follow this link.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"lv": {"name": "lv", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "lq": {"name": "lq", "group": "Ungrouped variables", "definition": "random(7 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(11 .. 14#1)", "description": "", "templateType": "randrange", "can_override": false}, "uq": {"name": "uq", "group": "Ungrouped variables", "definition": "random(15 .. 22#1)", "description": "", "templateType": "randrange", "can_override": false}, "hv": {"name": "hv", "group": "Ungrouped variables", "definition": "random(23 .. 30#1)", "description": "", "templateType": "randrange", "can_override": false}, "IQR": {"name": "IQR", "group": "Ungrouped variables", "definition": "uq-lq", "description": "", "templateType": "anything", "can_override": false}, "range": {"name": "range", "group": "Ungrouped variables", "definition": "hv-lv", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["lv", "lq", "m", "uq", "hv", "IQR", "range"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "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": "Which of these statements are true and which are false?
", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["The range of the data is $\\var{range}$.", "The Interquarttile range of the data is larger than the range of the data.", "You can calculate the mean of the data from this Box plot.", "The median of the data is $\\var{m}$.
", "The mode of the data is $\\var{lv-3}$."], "matrix": [["1", 0], [0, "1"], [0, "1"], ["1", 0], [0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["True.", "False."]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "SA6 Calculate Range", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Michael Pan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23528/"}], "tags": ["mean", "measures of average and spread", "median", "mode", "range", "taxonomy"], "metadata": {"description": "This question provides a list of data to the student. They are asked to find the \"range\".
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "A random sample of 20 residents from Newcastle were asked about the number of times they went to see a play at the theatre last year.
\nHere is the list of their answers:
\n| $\\var{a[0]}$ | \n$\\var{a[1]}$ | \n$\\var{a[2]}$ | \n$\\var{a[3]}$ | \n$\\var{a[4]}$ | \n$\\var{a[5]}$ | \n$\\var{a[6]}$ | \n$\\var{a[7]}$ | \n$\\var{a[8]}$ | \n$\\var{a[9]}$ | \n
| $\\var{a[10]}$ | \n$\\var{a[11]}$ | \n$\\var{a[12]}$ | \n$\\var{a[13]}$ | \n$\\var{a[14]}$ | \n$\\var{a[15]}$ | \n$\\var{a[16]}$ | \n$\\var{a[17]}$ | \n$\\var{a[18]}$ | \n$\\var{a[19]}$ | \n
Range is the difference between the highest and the lowest value in the data.
\nTo find this, we subtract the lowest value from the highest value:
\n\\[ \\var{max(a)} - \\var{min(a)} = \\var{range} \\text{.}\\]
\n\nUse this link to find some resources which will help you revise this topic.
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", "licence": "None specified"}, "statement": "Calculate the Mean from a list
", "advice": "The MEAN is the sum, divided by the number of values summed i.e.
$\\frac{\\var{list[0]} + \\var{list[1]} + \\var{list[2]} + \\var{list[3]} + \\var{list[4]}}{5}$
use your calculator to find
\nmean = $\\var{mean}$.
\n\nUse this link to find some resources which will help you revise this topic.
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{list}
Calculate the mean: [[0]]
This question provides a list of data to the student. They are asked to find the \"median\".
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "A random sample of 20 residents from Newcastle were asked about the number of times they went to see a play at the theatre last year.
\nHere is the list of their answers:
\n| $\\var{a[0]}$ | \n$\\var{a[1]}$ | \n$\\var{a[2]}$ | \n$\\var{a[3]}$ | \n$\\var{a[4]}$ | \n$\\var{a[5]}$ | \n$\\var{a[6]}$ | \n$\\var{a[7]}$ | \n$\\var{a[8]}$ | \n$\\var{a[9]}$ | \n
| $\\var{a[10]}$ | \n$\\var{a[11]}$ | \n$\\var{a[12]}$ | \n$\\var{a[13]}$ | \n$\\var{a[14]}$ | \n$\\var{a[15]}$ | \n$\\var{a[16]}$ | \n$\\var{a[17]}$ | \n$\\var{a[18]}$ | \n$\\var{a[19]}$ | \n
The median is the middle value. We need to sort the list in order:
\n\\[ \\var{a_s[0]}, \\quad \\var{a_s[1]}, \\quad \\var{a_s[2]}, \\quad \\var{a_s[3]}, \\quad \\var{a_s[4]}, \\quad \\var{a_s[5]}, \\quad \\var{a_s[6]}, \\quad \\var{a_s[7]}, \\quad \\var{a_s[8]}, \\quad \\var{a_s[9]}, \\quad \\var{a_s[10]}, \\quad \\var{a_s[11]}, \\quad \\var{a_s[12]}, \\quad \\var{a_s[13]}, \\quad \\var{a_s[14]}, \\quad \\var{a_s[15]}, \\quad \\var{a_s[16]}, \\quad \\var{a_s[17]}, \\quad \\var{a_s[18]}, \\quad \\var{a_s[19]} \\]
\nThere is an even number of responses, so there are two numbers in the middle (10th and 11th place). To find the median, we need to find the mean of these two numbers $\\var{a_s[9]}$ and $\\var{a_s[10]}$:
\n\\begin{align}
\\frac{\\var{a_s[9]} + \\var{a_s[10]}}{2} &= \\frac{\\var{a_s[9] + a_s[10]}}{2} \\\\
&= \\var{median} \\text{.}
\\end{align}
\n
Use this link to find some resources which will help you revise this topic.
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Match each of the following with what they measure.
", "advice": "The mean is a measure of location or central tendancy. It is calcuated by summing all of the data values and dividing by the number of values.
\nThe median is a measure of location or central tendancy. It is the middle value of an ordered data set.
\nThe inter-quartile range is a measure of spread. The interquartile range is the difference between upper and lower quartiles.The lower quartile, or first quartile (Q1), is the value under which 25% of data points are found when they are arranged in increasing order. The upper quartile, or third quartile (Q3), is the value under which 75% of data points are found when arranged in increasing order. The inter-quartile range therefore gives us an idea of the middle 50% of the ordered data set.
\nThe standard deviation is a measure of spread. It measures the dispersion of a data set relative to its mean.
\nThe variance is a measure spread because it is the square of the standard deviation.
\nA p-value the probability that a particular statistical measure, such as the mean or standard deviation, of an assumed probability distribution will be greater than or equal to (or less than or equal to in some instances) observed results. A p-value is used to determine statistical significance, not measures of spread or location.
\nUse this link to find some resources which will help you revise this topic.
", "rulesets": {"std": ["all", "fractionNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "function dragpoint_board() {\n var scope = question.scope;\n\n JXG.Options.text.display = 'internal';\n \n var yo0 = scope.variables.yo0.value;\n var yo1 = scope.variables.yo1.value;\n var yo2 = scope.variables.yo2.value;\n var yo3 = scope.variables.yo3.value;\n var yo4 = scope.variables.yo4.value;\n var yo5 = scope.variables.yo5.value;\n var yo6 = scope.variables.yo6.value;\n var yo7 = scope.variables.yo7.value; \n var yo8 = scope.variables.yo8.value;\n var yo9 = scope.variables.yo9.value; \n \n var div = Numbas.extensions.jsxgraph.makeBoard('550px','550px',{boundingBox:[-0.8,82,16,-8], axis:false, grid:true});\n \n $(question.display.html).find('#dragpoint').append(div);\n \n var board = div.board;\n \nboard.suspendUpdate(); \n\n \n var dataArr = [yo0,yo5,0,yo1,yo6,0,yo2,yo7,0,yo3,yo8,0,yo4,yo9]; \n \n var xaxis = board.create('axis', [[0, 0], [12, 0]], {withLabel: true, name: \"Bank\", label: {offset: [250,-30]}});\n \n xaxis.removeAllTicks(); \n \n board.create('axis', [[0, 0], [0, 10]], {hideTicks:true, withLabel: false, name: \"\", label: {offset: [-110,300]}});\n \n var pop0 = board.create('point', [1.5,0],{name:'Morgan',fixed:true,size:0,color:'black',face:'diamond', label:{offset:[-20,-8]}});\n var pop1 = board.create('point',[4.5,0],{name:'Strome',fixed:true,size:0,color:'black',face:'diamond', label:{offset:[-20,-8]}});\n var pop2 = board.create('point',[7.5,0],{name:'Bentley',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n var pop3 = board.create('point',[10.5,0],{name:'Sand',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n var pop4 = board.create('point',[13.5,0],{name:'Karchen',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n\n var leg1 = board.create('point',[12,75],{name:'last year',fixed:true,size:6,color:'#DA2228', face:'square', label:{offset:[9,0]}});\n var leg2 = board.create('point',[12,72],{name:'this year',fixed:true,size:6,color:'#6F1B75', face:'square', label:{offset:[9,0]}});\n \n \n// var chart = board.createElement('chart', dataArr, \n // {chartStyle:'bar', fillOpacity:1, width:1,\n // colorArray:['#8E1B77','#8E1B77','Red','Red','blue','red','blue','red','red','blue', 'red','blue','red','red'], shadow:false});\n \n//var chart = board.createElement('chart', dataArr, \n // {chartStyle:'bar', width:1,fillOpacity:1, fillColor:'red', shadow:false}); \n \n \n var a = board.create('chart', [[1,2,3],[yo0,yo5,0]], {chartStyle:'bar',colors:['#DA2228','#6F1B75','#6F1B75'],width:1,fillOpacity:1});\n var b = board.create('chart', [[4,5,6],[yo1,yo6,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n var c = board.create('chart', [[7,8,9],[yo2,yo7,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n var d = board.create('chart', [[10,11,12],[yo3,yo8,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n var e = board.create('chart', [[13,14],[yo4,yo9]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75'],fillOpacity:1});\n \n board.unsuspendUpdate();\n \n var txt1 = board.create('text',[-0.3,30, 'Investment \u00a3(m)'], {fontColor:'black', fontSize:14, rotate:90});\n \n // var txt = board.create('text',[0.5,75, 'Investment (m)'], {fontSize:14, rotate:90});\n \n // var txt1 = board.create('text',[8,76, 'red bars represent 2010'], {fontColor:'red', fontSize:14, rotate:90});\n \n // var txt2 = board.create('text',[8,73, 'blue bars represents 2011'], {fontSize:14, rotate:90});\n\n // var myColors = new Array('red', 'blue', 'white','red', 'blue', 'white','red', 'blue', 'white','red', 'blue', 'white','red', 'blue');\n \n \n \n //board.unsuspendUpdate();\n\n // Rotate text around the lower left corner (-2,-1) by 30 degrees.\n // var tRot = board.create('transform', [90.0*Math.PI/180.0, -1,40], {type:'rotate'}); \n // tRot.bindTo(txt);\n // board.update();\n\n \n//var chart2 = board.createElement('chart', dataArr, {chartStyle:'line,point'});\n//chart2[0].setProperty('strokeColor:black','strokeWidth:2','shadow:true');\n//for(var i=0; i<11;i++) {\n // chart2[1][i].setProperty({strokeColor:'black',fillColor:'white',face:'[]', size:4, strokeWidth:2});\n//}\n//board.unsuspendUpdate(); \n \n //board.unsuspendUpdate();\n\n}\n\nquestion.signals.on('HTMLAttached',function() {\n dragpoint_board();\n});", "css": "table#values th {\n background: none;\n text-align: center;\n}"}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Variance", "Mean", "Median", "Inter-quartile range", "P-value", "Standard deviation"], "matrix": [["1", 0, 0], [0, "1", 0], [0, "1", 0], ["1", 0, 0], [0, 0, "1"], ["1", 0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["Measure of Spread", "Measure of location (average)", "Neither measure of location nor measure of spread"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "SA13 Correlation", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Upuli Wickramaarachchi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23527/"}], "tags": [], "metadata": {"description": "Tests understanding of scatter plots and related concepts.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "The scatter plot below shows the relationship between an employee’s height in centimetres and how long it takes them to walk to work in minutes.
\n| time (mins) | \n{drawgraph()} | \n
| \n | height (cm) | \n
The graph shows that there is a positive correlation between a person's height and how long it takes them to walk to work.
\nA postive correlation is a relationship between two variables where both variables move in the same diection.
\nThis tells us that as a person's height increases, the time it takes to walk to work increases.
\nUse this link to find some resources which will help you revise this topic
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", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["In general, there is a positive correlation between a person's height and how long it takes them to walk to work.
", "In general, there is a negative correlation between a person's height and how long it takes them to walk to work.
", "In general, there is a no correlation between a person's height and how long it takes them to walk to work.
"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "SA14 Probability - \"sample space\"", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}], "tags": [], "metadata": {"description": "Calculate probability of selecting coloured counters from a bag.
", "licence": "None specified"}, "statement": "A bag contains:
$\\var{srn}$ small, red tokens,
$\\var{sbn}$ small, blue tokens,
$\\var{brn}$ large, red tokens, and
$\\var{bbn}$ large, blue tokens.
A probability is a fraction. You can give your answer as a fraction, decimal or percentage as these are all equivalent.
The formula for probability is:
\\[ P(A) = \\frac{\\text{number of possibilities for A}}{\\text{number of total possible outcomes}} \\]
\nFor this question the total possible outcomes are $\\var{srn}+\\var{sbn}+\\var{brn}+\\var{bbn} = \\var{total}$.
Therefore
\\[ P(\\text{A large red token}) = \\frac{\\var{brn}}{\\var{total}} = \\var[fractionnumbers]{brn/total}\\]
\nFor this question we need to know the total number of small tokens, i.e. $\\var{srn}+\\var{sbn} = \\var{srn+sbn}$.
Therefore
\\[ P(\\text{A small token}) = \\frac{\\var{srn+sbn}}{\\var{total}} = \\var[fractionnumbers]{(srn+sbn)/total}\\]
\n\nUse this link to find some resources which will help you revise this topic.
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What is the probability that it is a large, red token?
Give your answer as a fraction, or a decimal correct to 2dp.
You take a token at random.
What is the probability that it is a small token?
Give your answer as a fraction, or a decimal correct to 2dp.
This is a tool for you! It is here to help you diagnose whether there are any maths or statistics pre-requisites for your course that you may want to brush up on. If at any point you are struggling with any question you should find a link at the end of the \"reveal answer\" section that will take you to some recommended online resources on that subject area. You can also always contact the Maths and Stats Help team (MaSH) to arrange a one to one appointment or check out our workshop timetable to see if you can access the support you need that way. Find all this information via our website here!
", "end_message": "Thanks for completing the Skills Audit. You can attempt this as many times as you need. Remember the score is not what matters - this is in no way assessed work - this is simply a tool for working out whether you may need to brush up on anything to ensure that you can access all the material on your course and get off to the best possible start.
\nDon't forget to look up what support is available to you through our web pages here!
", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "inreview"}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "type": "exam", "contributors": [{"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "extensions": ["chemistry", "eukleides", "geogebra", "jsxgraph", "codewords", "permutations", "polynomials", "quantities", "random_person", "stats", "visjs"], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "The answer is a comma-separated list of numbers.
\nThe list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.
\nYou can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.
", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n if(settings[\"correctAnswerFractions\"],\n map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n ,\n settings[\"correctAnswer\"]\n ),\n settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n \"Enter a list of numbers separated by{settings['separator']}.\",\n \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n true,\n let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n warn(wrong+\" is not a valid number\");\n fail(wrong+\" is not a valid number.\")\n )\n )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n multiply_credit(0.5,\"Not in order\")\n )\n\nincluded:\nmap(\n let(\n num_student,len(filter(x=y,y,interpreted_answer)),\n num_expected,len(filter(x=y,y,expected_numbers)),\n switch(\n num_student=num_expected,\n true,\n num_studentIs every number in the student's list valid?
", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n true,\n let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n warn(wrong+\" is not a valid number\");\n fail(wrong+\" is not a valid number.\")\n )\n )"}, {"name": "is_sorted", "description": "Are the student's answers in ascending order?
", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n multiply_credit(0.5,\"Not in order\")\n )"}, {"name": "included", "description": "Is each number in the expected answer present in the student's list the correct number of times?
", "definition": "map(\n let(\n num_student,len(filter(x=y,y,interpreted_answer)),\n num_expected,len(filter(x=y,y,expected_numbers)),\n switch(\n num_student=num_expected,\n true,\n num_studentTrue if the student's list doesn't contain any numbers that aren't in the expected answer.
", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n true\n ,\n incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n false\n )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n map(\n if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n ,x\n ,bits\n )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "Should the answer be considered as a set, so the number of times an element occurs doesn't matter?
", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "Numbers included in the student's answer that are not in the expected list.
", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}, {"source": {"pk": 28, "author": {"name": "Marie Nicholson", "pk": 1799}, "edit_page": "/part_type/28/edit"}, "name": "True/False", "short_name": "true-false", "description": "The answer is either 'True' or 'False'
", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["True", "False"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"True\"", "help_url": "", "hint": "", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [["question-resources/Picture1_caMIdF1.png", "/srv/numbas/media/question-resources/Picture1_caMIdF1.png"], ["question-resources/Picture2_6KE4ZpW.png", "/srv/numbas/media/question-resources/Picture2_6KE4ZpW.png"], ["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"]]}