Questions on rearranging expressions, expanding brackets and collecting like terms.
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", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["d", "f", "g", "h", "x", "gcd_hfdg", "hf_coprime", "dg_coprime", "finalb"], "type": "question", "rulesets": {}, "variable_groups": [], "statement": "", "advice": "We are asked to solve the equation
\n\\[ \\var{d}x-\\var{f}=\\var{g}x+\\var{h} \\]
\nIn this equation, there are $x$ terms and constant terms on both sides of the equals sign.
\nTo solve this equation, we must rearrange it to get $x$ on its own.
\n\\begin{align}
\\var{d}x-\\var{f} &= \\var{g}x+\\var{h} \\\\[0.5em]
\\var{d}x-\\var{g}x &= \\var{h}+\\var{f} & \\text{Move } x \\text{ terms to the left, and constant terms to the right.}\\\\[0.5em]
\\simplify{{d-g}*x} &= {\\var{h+f}} & \\text{Collect like terms together.}\\\\[0.5em]
x &=\\frac{\\var{h+f}}{\\var{d-g}} & \\text{Divide both sides by } \\var{d-g} \\text{.} \\\\[0.5em]
x &= \\simplify{{h+f}/{d-g}}
\\end{align}
$\\var{d}x-\\var{f}=\\var{g}x+\\var{h}$
\nWhat is the value of $x$?
\n$x = $ [[0]]
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\nThe answer is always an integer.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": [], "type": "question", "rulesets": {}, "advice": "We need to solve the equation
\n\\[ \\var{a}x+\\var{b}=\\var{c} \\]
\nTo solve this equation, we must rearrange the equation to put $x$ on its own.
\nTo do this, we should subtract $\\var{b}$ from both sides and then divide through by $\\var{a}$ to get the value for $x$.
\n\\begin{align}
\\var{a}x+\\var{b}&=\\var{c} \\\\[0.5em]
\\var{a}x&=\\var{c}-\\var{b} & \\text{Subtract } \\var{b} \\text{ from both sides} \\\\[0.5em]
\\var{a}x&=\\var{c-b} \\\\[0.5em]
x&=\\frac{\\var{c-b}}{\\var{a}} & \\text{Divide both sides by } \\var{a} \\\\[0.5em]
x&=\\simplify{{c-b}/{a}}
\\end{align}
$\\var{a}x+\\var{b}=\\var{c}$
\nWhat is the value of $x$?
\n$x = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Expand the expressions below by multiplying each of the terms inside the brackets by the term outside. Give each 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[1]}({a[2]}x+{a[3]})}&=
\\simplify[!collectNumbers]{({a[1]}{a[2]})x+({a[1]}{a[3]})}\\\\&
=\\simplify{{a[1]}*{a[2]}x+{a[1]}{a[3]}}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[4]}({a[5]}x+{a[6]})}&=
\\simplify[!collectNumbers]{{a[4]}{a[5]}x+{a[4]}{a[6]}}\\\\&=
\\simplify{{a[4]}*{a[5]}x+{a[4]}{a[6]}}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[7]}({a[8]}x^2+{a[9]}y)}&=
\\simplify[!collectNumbers]{{a[7]}{a[8]}x^2+{a[7]}{a[9]}y}\\\\&=
\\simplify{{a[7]}*{a[8]}x^2+{a[7]}*{a[9]}y}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[10]}({a[11]}x^2+{a[12]}y)}&=
\\simplify[!collectNumbers]{{a[10]}{a[11]}x^2+{a[10]}{a[12]}y}\\\\&=
\\simplify{{a[10]}*{a[11]}x^2+{a[10]}*{a[12]}y}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[13]}x({a[14]}x^2+{a[15]}x+{a[16]})}&=
\\simplify[!collectNumbers]{{a[13]}x{a[14]}x^2+{a[13]}x{a[15]}x+{a[13]}x{a[16]}}\\\\&=
\\simplify{{a[13]}{a[14]}x^3+{a[13]}{a[15]}x^2+{a[13]}{a[16]}x}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[17]}x({a[18]}x^2+{a[19]}x+{a[20]})}&=
\\simplify[!collectNumbers]{{a[17]}x{a[18]}x^2+{a[17]}x{a[19]}x+{a[17]}x{a[20]}}\\\\&=
\\simplify{{a[17]}{a[18]}x^3+{a[17]}{a[19]}x^2+{a[17]}{a[20]}x}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[21]}x({a[22]}x^2+{a[23]}x)+{a[24]}x^2+{a[25]}x^3}&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]})+x^2{a[24]}+x^3({a[21]}{a[22]})+x^3{a[25]}}\\\\&=
\\simplify[!collectNumbers]{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\\\&=
\\simplify{x^2({a[21]}{a[23]}+{a[24]})+x^3({a[21]}{a[22]}+{a[25]})}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{({a[26]}x^2+{a[27]}x^3)+{a[28]}x({a[29]}x^2+{a[30]}x)}&=
\\simplify[!collectNumbers]{x^2({a[26]})+x^2({a[28]}{a[30]})+x^3({a[28]}{a[29]})+x^3({a[27]})}\\\\&=
\\simplify[!collectNumbers]{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\\\&=
\\simplify{x^2({a[26]}+{a[28]}{a[30]})+x^3({a[28]}{a[29]}+{a[27]})}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[31]}({a[32]}x+{a[33]}y)+{a[34]}x({a[42]}+{a[35]}y)}&=
\\simplify[!collectNumbers]{({a[31]}{a[32]})x+({a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify[!collectNumbers]{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\\\&=
\\simplify{({a[31]}{a[32]}+{a[34]}{a[42]})x+{a[31]}{a[33]}y+{a[34]}{a[35]}x*y}\\text{.}
\\end{align}
\\]
\\[
\\begin{align}
\\simplify[terms]{{a[36]}a^2({a[37]}+{a[38]}b)+{a[39]}b^2({a[40]}a+{a[41]}b)}&=
\\simplify[!collectNumbers]{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\\\&=
\\simplify{{a[37]}{a[36]}a^2+{a[38]}{a[36]}a^2b+{a[40]}{a[39]}a*b^2+{a[39]}{a[41]}b^3}\\text{.}
\\end{align}
\\]
$\\simplify{{a[1]}({a[2]}x+{a[3]})}=$ [[0]]
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", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{a[37]*a[36]}a^2+{a[38]*a[36]}a^2b+{a[40]*a[39]}a*b^2+{a[39]*a[41]}b^3", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "mustmatchpattern": {"pattern": "`+-$n`?*a^2 + `+-$n`?*a^2*b + `+-$n`?*a*b^2 + `+-$n`?*b^3", "partialCredit": 0, "message": "It doesn't look like you've expanded - make sure you don't use any brackets in your answer.
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\nBoth terms have a common factor of $2$.
\n\\begin{align}
\\simplify{2{a[0]}x+2{b[0]}}&=
(\\simplify[]{2{a[0]}})x+2\\times\\var{b[0]}\\\\
&=\\simplify[]{2({a[0]}x+{b[0]})}
\\end{align}
Both terms have common factors of $6$ and $y$.
\n\\begin{align}
\\simplify{6{a[1]}y+6{b[1]}y^2}&= 6 \\times \\var{a[1]} y + 6 \\times \\var{b[1]} y^2 \\\\
&= 6 \\times (\\simplify{{a[1]}y + {b[1]}y^2}) \\\\
&=6y(\\simplify[]{{a[1]}+{b[1]}y})
\\end{align}
Both terms have common factors of $x$, $y$ and $z$.
\n\\begin{align}
\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}&=\\var{a[2]} \\times xyz + \\var{b[2]} \\times xyz \\times xyz\\\\
&=xyz(\\var{a[2]} + \\var{b[2]} xyz)
\\end{align}
All three terms have a common factor of $5$.
\n\\begin{align}
\\simplify{5{a[3]}d+5{b[3]}r+5m}&= 5 \\times \\var{a[3]} d+5 \\times \\var{b[3]} r + 5 m \\\\
&=\\simplify[]{5({a[3]}d+{b[3]}r+m)}
\\end{align}
All the terms have common factors of $6$, $c$ and $d$.
\n\\begin{align}
\\simplify{6{a[4]}cd^2+6{b[4]}c^2d+6{c[1]}c^2d^2} &= 6 \\times \\var{a[4]} c d^2 \\;+\\; 6 \\times \\var{b[4]} c^2 d \\;+\\; 6 \\times \\var{c[1]} c^2 d^2 \\\\
&= 6(\\var{a[4]} c d^2 + \\var{b[4]} c^2 d + \\var{c[1]} c^2 d^2) \\\\
&=6cd(\\var{a[4]}d+\\var{b[4]}c+\\var{c[1]}cd)
\\end{align}
An expression can be factorised by finding common factors of each term in the expression.
\nCompletely factorise the following expressions by finding their common factors.
\nMake sure that you include a multiplication symbol * between each algebraic variable, and before brackets, e.g. a*b*(x+1) instead of ab(x+1). Otherwise, the system might not accept your answer.
Vector of every other random prime number
", "group": "Ungrouped variables", "definition": "repeat(random([3, 11, 17, 29, 37, 43] except b),50)"}, "x2": {"templateType": "anything", "name": "x2", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x3": {"templateType": "anything", "name": "x3", "description": "", "group": "Ungrouped variables", "definition": "random(1..5 except [x2])"}, "b": {"templateType": "anything", "name": "b", "description": "Vector of the other every other random prime number
", "group": "Ungrouped variables", "definition": "repeat(random(2, 7, 13, 23, 31, 41, 53),50)"}, "c": {"templateType": "anything", "name": "c", "description": "extra primes for when you need a third constant
", "group": "Ungrouped variables", "definition": "repeat(random([ 5, 19, 47] ),50)"}, "x1": {"templateType": "anything", "name": "x1", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x4": {"templateType": "anything", "name": "x4", "description": "", "group": "Ungrouped variables", "definition": "random(-5..-1)\n\n"}, "x5": {"templateType": "anything", "name": "x5", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}, "x6": {"templateType": "anything", "name": "x6", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)"}}, "tags": ["common factors", "common factors of linear algebraic equations", "common factors of quadratic equations", "finding common factors", "Linear equations", "linear equations", "quadratic equations", "Quadratic Equations", "Quadratic equations", "taxonomy"], "ungrouped_variables": ["a", "b", "c", "x1", "x2", "x3", "x4", "x5", "x6", "x7"], "functions": {}, "metadata": {"description": "Factorise polynomials by identifying common factors. The first expression has a constant common factor; the rest have common factors involving variables.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [], "parts": [{"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "2({a[0]}x+{b[0]})", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "$\\simplify{{2*a[0]}x+{2*b[0]}}=$ [[0]]
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", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "x*y*z*({a[2]}+{b[2]}x*y*z)", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "$\\simplify{{a[2]}x*y*z+{b[2]}x^2y^2z^2}=$ [[0]]
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", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "variableReplacements": [], "gaps": [{"variableReplacements": [], "answer": "6c*d*({a[4]}d+{b[4]}c+{c[1]}c*d)", "showPreview": true, "expectedVariableNames": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "question.mark_factorised(this);"}}, "showCorrectAnswer": true, "failureRate": 1, "type": "jme", "checkVariableNames": false, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "marks": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "prompt": "$\\simplify{6{a[4]}c*d^2+6{b[4]}c^2d+6{c[1]}c^2d^2}=$ [[0]]
", "customMarkingAlgorithm": "", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": "1000"}, "rulesets": {}}, {"name": "Create an algebraic expression from a word problem, simplify, and evaluate", "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": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "tags": ["algebraic expressions", "collect terms", "create algebraic expressions", "simplify algebraic expressions", "simplifying algebraic expressions", "taxonomy"], "metadata": {"description": "Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.
\nThe word problem is about the costs of sweets in a sweet shop.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{pname} eats a lot of sweets. You are trying to work out the cost of the sweets that {pname} ate last week.
\n{pname} ate $\\var{a1}$ packets of lollipops, $\\var{b1}$ packets of toffee and $\\simplify{{c1}}$ packets of jelly sweets.
\nYou know that a packet of toffee costs $£1$ more than a packet of lollipops, and a packet of jelly sweets costs half as much as a packet of toffees.
", "advice": "We are told that the price of a packet of lollipops is represented by the letter $x$.
\nA packet of toffee costs $£1$ more than a packet of lollipops, i.e. $x+1$.
\nA packet of jelly sweets costs half as much as a packet of toffee, so $\\frac{1}{2}(x+1)$.
\nTo find the total cost, multiply the expressions above for the cost of each kind of sweet by the number of packets eaten, and add them together.
\nWithout simplifying, we obtain:
\n\\begin{align}
\\text{Cost} &= \\simplify[]{{a1}x+{b1}(x+1) + {c1}*(1/2)*(x+1)} \\\\
&= \\simplify[]{{a1}x+{b1}(x+1) + {c1/2}*(x+1)}
\\text{.}
\\end{align}
The first step in simplifying this expression is to expand both sets of brackets:
\n\\begin{align}
\\simplify[]{ {a1}x + {b1}(x+1) + {c1/2}*(x+1)} &= \\simplify[]{ {a1}x + {b1}x + {b1}*1 + {c1/2}x + {c1/2}*1} \\\\
&= \\simplify[] { {a1}x + {b1}x + {b1} + {c1/2}x + {c1/2} } \\text{.}
\\end{align}
Finally, collect like terms:
\n\\begin{align}
\\simplify[] { {a1}x + {b1}x + {b1} + {c1/2}x + {c1/2} } &= \\simplify[]{ {a1+b1+c1/2}x + {b1+c1/2} } \\text{.}
\\end{align}
Once we know that the price of a packet of lollipops is $£2$, we can substitute this for $x$ in the equation above.
\n\\begin{align}
\\text{Cost}&=\\simplify{ {a1+b1+c1/2}x+{b1+c1/2} }\\\\
&=\\var{a1+b1+c1/2} \\times 2+\\var{b1+c1/2} \\\\
&=\\var{(a1+b1+c1/2)*2+b1+c1/2} \\text{.}
\\end{align}
So {pname} spent $£\\var{total}$ on sweets last week.
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\nWrite an expression in terms of $x$ for the cost of each kind of sweet:
\nLollipops: £[[0]]
\nToffees: £[[1]]
\nJelly sweets: £[[2]]
Write an algebraic expression for the overall cost of the sweets {pname} ate, in terms of $x$.
\n£[[0]]
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\n£[[0]]
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\nCalculate {pname}'s total expenditure on sweets last week.
\n£[[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": "({a1}+{b1}+{c1}/2)2+{b1}+{c1}/2", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["(", ")"], "showStrings": true, "partialCredit": 0, "message": "Don't use brackets
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\n$a_n =$ [[0]]
\n"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "a1+(small-1)*d", "maxValue": "a1+(small-1)*d", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "\n
Find the $\\var{small}^{\\text{th}}$ term
\n$a_{\\var{small}} = $ [[0]]
\n"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": false, "scripts": {}, "minValue": "a1+(large-1)*d", "maxValue": "a1+(large-1)*d", "marks": 1, "variableReplacements": []}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "prompt": "
Find the $\\var{large}^{\\text{th}}$ term
\n$a_{\\var{large}} = $[[0]]
"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.
"}, "tags": ["arithmetic sequences", "nth term", "sequences", "taxonomy"], "variables": {"large": {"templateType": "anything", "description": "A large index to compute
", "definition": "random(10..50#5)*10", "name": "large", "group": "Ungrouped variables"}, "small": {"templateType": "anything", "description": "A small index to compute
", "definition": "random(6..10)", "name": "small", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "description": "The first term in the sequence
", "definition": "random(1..90)", "name": "a1", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "description": "", "definition": "random(3..13 except 10)", "name": "d", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "d", "small", "large"], "statement": "In this question, consider the sequence
\n\\[ a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \\]
\nA helpful person has drawn out a table of the terms so far.
\n| $\\boldsymbol{n}$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n$\\ldots$ | \n
|---|---|---|---|---|---|
| $\\boldsymbol{a_n}$ | \n$\\var{a1}$ | \n$\\var{a1+d}$ | \n$\\var{a1+2d}$ | \n$\\var{a1+3d}$ | \n$\\ldots$ | \n
The formula for the $n^\\text{th}$ term, $a_n$, of an arithmetic sequence is
\n\\[ a_n=a_1+(n-1)d \\text{.} \\]
\n$a_1$ is the first term, and $d$ is the common difference between adjacent terms.
\nIn the given sequence, the common difference is $\\var{a1+d} - \\var{a1} = \\var{d}$, and the first term is $\\var{a1}$.
\nSo, the formula for this sequence is
\n\\[ a_n = \\var{a1} + (n-1) \\times \\var{d} \\text{.} \\]
\n\\[ a_\\var{small} = \\var{a1} + (\\var{small}-1) \\times \\var{d} = \\var{a1+(small-1)*d} \\text{.} \\]
\n\\[ a_\\var{large} = \\var{a1} + (\\var{large}-1) \\times \\var{d} = \\var{a1+(large-1)*d} \\text{.} \\]
\nrandom variables for part 1
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$\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}=$ [[0]]
You must condense your answer to fully simplify.
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", "unitTests": [], "customName": ""}, {"extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "checkVariableNames": false, "useCustomName": false, "mustmatchpattern": {"nameToCompare": "", "partialCredit": 0, "pattern": "$n*x+$n*y", "message": "You haven't simplified: you still have two or more like terms that should be collected together."}, "unitTests": [], "answerSimplification": "all", "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "failureRate": 1, "variableReplacements": [], "vsetRange": [0, 1], "maxlength": {"partialCredit": 0, "message": "You must condense your answer to fully simplify. *'s are not needed to indicate multiplication here.
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", "unitTests": [], "customName": ""}], "advice": "When simplifying expressions, only terms of the same type or like terms can be added together.
\nAlgebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.
\n\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]
\\[
\\begin{align}
\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\
&=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]}
\\end{align}
\\]
\\[
\\begin{align}
\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\
&=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5
\\end{align}
\\]
\\[
\\begin{align}
\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc
&=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\
&=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c
\\end{align}
\\]
\\[
\\begin{align}
\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2
&=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\
&=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab
\\end{align}
\\]
\\[
\\begin{align}
\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y
&=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\
&=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\
&=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y
\\end{align}
\\]
\\[
\\begin{align}
\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2
&=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z
\\end{align}
\\]
\\[
\\begin{align}
\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)
&= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\
&= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\
&= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y}
\\end{align}
\\]
For each expression below, collect like terms and expand brackets.
\nThe * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.
Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.
"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Substitute values into formulas", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "metadata": {"description": "Substitute given values into formulas.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["r", "x1", "n", "x2", "const", "sales"], "type": "question", "advice": "When inserting numbers into your calculator make sure you place brackets correctly.
\nAs $x = \\var{n+2}$,
\nsubstitute $\\var{n+2}$ into $\\var{x2}x^2 + \\var{x1}x + \\var{const}$.
\n\\begin{align}
\\var{x2}x^2 + \\var{x1}x + \\var{const} &= \\var{x2} (\\var{n+2})^2 + \\var{x1}(\\var{n+2}) + \\var{const} \\\\
&= \\simplify{{x2} ({n+2})^2 + {x1}({n+2}) + {const}}\\,.
\\end{align}
b)
\nAs $y = \\var{n}$,
\nsubstitute $\\var{n}$ into $\\var{n+1}y^2-\\var{x2}y$.
\n\\begin{align}
\\var{n+1}y^2-\\var{x2}y &= \\var{n+1}(\\var{n})^2-\\var{x2}(\\var{n}) \\\\
&= \\simplify{{n+1}({n})^2-{x2}({n})}\\,.
\\end{align}
c)
As we are given a temperature in degrees Celcius, $T_C = \\var{T_C}°C.$
\nSubstituting $T_C$ into $T_C = 1.8\\,T_C + 32$.
\n\\begin{align}
T_F &=1.8\\, T_C+32 \\\\
&=1.8 (\\var{T_C}) + 32 \\\\
&= \\var{dpformat(1.8 {T_C} +32, 1)}\\,°F\\,.
\\end{align}
Substitute the given values in the equations below.
", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{x2}{n+2}^2+{x1}{n+2}+{const}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "A curve is defined by a function $y=\\simplify{{x2}x^2 + {x1}x + {const}}$.
\nWhat is the $y$ coordinate value of the point on the curve at $x=\\var{n+2}$?
\n$y =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{n+1}{n}^2-{x2}{n}", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{n+1}{n}^2-{x2}{n}", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "{name[n]} sells luxury yachts.
\nThe predicted sales of the luxury yachts are defined by
\n\\[S=\\simplify{{n+1}y^2-{x2}y},\\]
\nwhere
$S$ is the number of sales predicted this year;
$y$ is the number of luxury yachts sold in the previous year.
{pronoun} sold {n} yachts in the previous year.
\nCalculate $S$, the number of sales predicted this year.
\n$S =$ [[0]]
"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "allowFractions": false, "minValue": "T_F", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "T_F", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "You can convert temperatures from degrees celsius to degrees fahrenheit by using the formula
\n\\[T_F=1.8\\, T_C+32,\\]
\nwhere
$T_F$ = Temperature in $°F$
$T_C$ = Temperature in $°C$.
Convert $\\var{T_C}°C$ into degrees fahrenheit.
\n$T_F =$ [[0]] $°F$
"}], "tags": ["predicted value", "substitution", "Substitution", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"pronoun": {"description": "Defines the pronoun in the question.
", "definition": "if(mod(n,2)=0,\"He\",\"She\")", "group": "Name variables", "name": "pronoun", "templateType": "anything"}, "T_C": {"description": "Creates a random integer value for the temperature in degrees celcius.
", "definition": "random(5..30#1)", "group": "Temperature conversion", "name": "T_C", "templateType": "anything"}, "name": {"description": "List of names to randomise. Can change to any name inserted
", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "group": "Name variables", "name": "name", "templateType": "anything"}, "n": {"description": "n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.
", "definition": "random(0..4#1)", "group": "Ungrouped variables", "name": "n", "templateType": "anything"}, "sales": {"description": "", "definition": "(n+1)n^2-x2*n", "group": "Ungrouped variables", "name": "sales", "templateType": "anything"}, "const": {"description": "The constant coefficient
", "definition": "random(1..100#1)", "group": "Ungrouped variables", "name": "const", "templateType": "anything"}, "T_F": {"description": "Creates a value for Temperature in fahrenheit.
", "definition": "T_C*1.8+32", "group": "Temperature conversion", "name": "T_F", "templateType": "anything"}, "r": {"description": "A random variable which will be inputted by the student.
", "definition": "random(1..50#0.1)", "group": "Ungrouped variables", "name": "r", "templateType": "anything"}, "x2": {"description": "The x^2 coefficient
", "definition": "random(1..(n+1)*n)", "group": "Ungrouped variables", "name": "x2", "templateType": "anything"}, "name2": {"description": "List of names to randomise. Can change to any name inserted
", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "group": "Name variables", "name": "name2", "templateType": "anything"}, "x1": {"description": "The x coefficient
", "definition": "random(1..50)", "group": "Ungrouped variables", "name": "x1", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}]}], "showstudentname": true, "percentPass": 0, "duration": 0, "type": "exam", "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}], "extensions": ["geogebra", "stats"], "custom_part_types": [], "resources": []}