Collecting like terms, expanding a bracket (distributive law), substitution of values, finding factors, simplifying algebraic fractions.
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", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0])[0..3]", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0])[0..2]", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..2]", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..2]", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..3]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0,0,0])[0..3]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "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": "$\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}$ = [[0]]
", "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": "Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x$ and $-2y$ have different variable parts and are therefore unlike terms (or not like terms).
\nWe can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x+3y$ equals $5x$ or $5y$! We can, however, say $2a+3a=5a$.
\nIn our question we look at all the terms with a variable part of $x$ and add up all the corresponding coefficients, we do the same for the $y$ terms and the $z$ terms:
\n\\[\\begin{align}
&\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}\\\\
&=\\simplify[basic]{({a[1]}+{a[2]}+{a[0]})x+({b[1]}+{b[2]}+{b[0]})y+({c[1]}+{c[2]}+{c[0]})z}\\end{align}\\]
We present this as the sum of three unlike terms:
\n\\[\\simplify{{sum(a)}x+{sum(b)}y+{sum(c)}z}\\]
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", "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": "Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x^2$ and $-2x$ have different variable parts and are therefore unlike terms (or not like terms).
\nWe can only collect like terms! Just like we can't say 2 m + 3 cm equals 5 m or 5 cm, we can't say $2x^2+3x$ equals $5x^2$ or $5x$! We can, however, say $2x^2+3x^2=5x^2$.
\nIn our question we look at all the terms with a variable part of $x^2$ and add up all the corresponding coefficients (the numbers in front of the variables), we do the same for the $x$ terms and the constant terms (the terms with no variable part):
\n\\[\\begin{align}&\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}\\\\&=\\simplify[basic]{({d[1]}+{d[0]})x^2+({f[1]}+{f[0]})x+({g[1]}+{g[0]})}\\end{align}\\]
\nWe present this as the sum of three unlike terms:
\n\\[\\simplify[!noleadingminus, basic]{{sum(d)}x^2+{sum(f)}x+{sum(g)}}\\]
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\n[[0]]$a$ + [[1]]
", "stepsPenalty": "2", "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": "The number in front of the bracket is multiplying the bracketed term, that is, each term in the brackets. Also, recall that a negative multiplied by a negative is a positive.
\n$\\begin{align*}
\\simplify{{nmult}({nxcoeff}a+{nconstant})}&=\\simplify[!noleadingminus]{{nmult}*{nxcoeff}a+{nmult} * {nconstant}}\\\\&=\\simplify[!noLeadingMinus]{{nmult*nxcoeff}a+{nmult*nconstant}}
\\end{align*}$
significant figures of velocity
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\n", "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": "
Be mindful of the order of operations and negatives when evaluating expressions.
\n\n
Substituting $a=\\var{aval}$, $b=\\var{bval}$ and $c=\\var{cval}$ into $b^2-4ac$ gives
\n\\[(\\var{bval})^2-4(\\var{aval})(\\var{cval})=\\simplify[basic]{{bval^2}-4{aval}{cval}}=\\simplify[basic]{{bval^2}+{-4*aval*cval}}=\\var{disans}.\\]
\nSubstituting $x=\\var{xval}$ and $y=\\var{yval}$ into $\\simplify{(x-{aval})^2+(y-{cval})^2}$ gives
\n\\[\\simplify[basic]{({xval}-{aval})^2+({yval}-{cval})^2}=(\\var{xval-aval})^2+(\\var{yval-cval})^2=\\var{(xval-aval)^2}+\\var{(yval-cval)^2}=\\var{cirans}.\\]
\nSubstituting $m_0=\\var{m0}$, $v=\\var{v_sig_figs}\\times 10^8$ and $c=3\\times 10^8$ into $m=\\dfrac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}}$ gives
\n\\[m=\\dfrac{\\var{m0}}{\\sqrt{1-\\frac{(\\var{v_sig_figs}\\times 10^8)^2}{(3 \\times 10^8)^2}}}=\\var{mans} \\quad\\text{ (to two decimal places).}\\]
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", "stepsPenalty": "10", "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": "Factors are things that multiply to make a product.
\nConsider the product $\\simplify{{prime^primepower}x^{xpower}y^{ypower}(x+{xconstant})^{xconstantpower} (z+{failconstant})^{failsafepower}}$.
\nWe can write this product as
\n$\\simplify[basic, alwaystimes]{{expanded}}$
\nAny combination of the above factors will still be a factor (since we can rearrange the product so that our collection of factors are all together and we can treat that collection as a single factor).
"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": "choices", "matrix": "gmarks"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Suppose that $\\var{factor1}${factor2} is a factor of an expression. What can be said of $-\\var{factor1}${factor2}?
", "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": "Since $-1$ can divide every number (just like $1$ can), if a positive number is a factor, then so is the negative of it.
\nIn particular, if $\\var{factor1}${factor2} is a factor of an expression, then that expression can be written as a product involving $\\var{factor1}${factor2}, for instance $\\var{factor1}${factor2}$w$ where $w$ is the other factor. Notice then that we can write the expression as $-\\var{factor1}${factor2}$(-w)$, that is, $-\\var{factor1}${factor2} is also a factor.
"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["It is also a factor.
", "It is not necessarily a factor.
", "It is definitely not a factor.
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\n\nWhat has {FirstName[0][0]} done to the first fraction in order to get the second? {FirstName[0][1]} has divided the top and bottom by [[0]] .
", "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": "The two terms in the numerator have a common factor of $\\var{amult}$. The two terms in the denominator also have a common factor of $\\var{amult}$. {FirstName[0][1]} has divided the numerator and denominator by the common factor of $\\var{amult}$ to get the equivalent fraction.
\n"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "amult", "maxValue": "amult", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "{FirstName[1][0]} has written $\\displaystyle\\frac{\\simplify{((x+{ar})/({f}x))}}{\\simplify{({br}y+{dr})}}$ in the equivalent form $\\displaystyle{\\simplify{(x+{ar})/({f*br}x*y+{f*dr}x)}}$.
\n\nWhat has {FirstName[1][0]} done to the first fraction in order to get the second? {FirstName[1][1]} has multiplied the top and bottom by [[0]] .
", "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": "The numerator and denominator have both become $\\var{f}x$ times larger. {FirstName[1][1]} has multiplied the numerator and denominator by $\\var{f}x$ to get an equivalent fraction.
"}], "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": "{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, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Which of the following are equivalent to $\\displaystyle{\\simplify{({g}x+{h})/(x+{h})}}$?
", "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": "When you have a fraction, $\\displaystyle{\\simplify{({g}x+{h})/(x+{h})}}$, it is equal to the sum of each term in the numerator being divided by the entire denominator. That is,
\n\n\\[\\displaystyle{\\simplify{({g}x+{h})/(x+{h})}}=\\simplify[!collectnumbers]{({g}x)/(x+{h})+({h})/(x+{h})}\\]
"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\var{g}$
", "$\\simplify{{g-1}x+{h}}$
", "$\\displaystyle\\simplify[!collectnumbers]{({g}x)/(x+{h})+({h})/(x+{h})}$
"], "matrix": [0, 0, "1"], "distractors": ["What common factor did you use?", "What common factor did you use?", ""]}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\displaystyle{\\simplify{({j}x+{k})/({l}y)}}$ is equal to:
", "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": "When you have a fraction, $\\displaystyle{\\simplify{({j}x+{k})/({l}y)}}$, it is equal to the sum of each term in the numerator being divided by the entire denominator. That is,
\n\n\\begin{align}\\displaystyle\\simplify{({j}x+{k})/({l}y)}&=\\simplify[!simplifyfractions]{{j}x/({l}y)+{k}/({l}y)}\\\\&=\\simplify{x/(2y)+{k}/({l}y)}\\end{align}
\nNote: In the last line we cancelled a common factor of $\\var{j}$ in the fraction $\\simplify[!simplifyfractions]{{j}x/({l}y)}$ to get the equivalent fraction $\\simplify{x/(2y)}$.
\n"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\displaystyle{\\simplify{(x+{k})/(2y)}}$
", "$\\displaystyle{\\simplify{x/(2y)+{k}}}$
", "$\\displaystyle{\\simplify{x/(2y)+{k}/({l}y)}}$
"], "matrix": [0, 0, "1"], "distractors": ["What common factor did you use?", "What common factor did you use?", ""]}, {"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": "Please fill in the gap to simplify the fraction on the left.
\n| $x+\\var{m}$ | \n= | \n[[0]] | \n
| $(x+\\var{m})(x+\\var{n})$ | \n\n | $x+\\var{n}$ | \n
These are equivalent fractions so the same number that multiplied/divided the denominator must also multiply/divide the numerator. Either of the following approaches could be taken:
\nThe expression $\\displaystyle{\\frac{\\var{p}z}{\\frac{\\var{p}}{\\var{p}z}}}$ can be simplified to [[0]] .
\nWhenever we have such a 'fraction on a fraction' we want to rewrite the fraction to it is just one fraction. We can do this by multiplying the numerator and denominator by the denominator of the smaller/inner fraction.
\nIn our case, the denominator of the smaller/fraction is $\\var{p}z$:
\n$\\begin{align}\\displaystyle \\frac{\\var{p}z}{\\frac{\\var{p}}{\\var{p}z}}&=\\frac{\\var{p}z\\times \\var{p}z}{\\frac{\\var{p}}{\\var{p}z}\\times\\var{p}z}\\\\&=\\frac{\\var{p}^2z^2}{\\var{p}}\\\\&=\\var{p}z^2.\\end{align}$
\n\nAlternatively, we can think of this as division by a fraction, and we can deal with that using multiplication by the reciprocal:
\n$\\begin{align}
\\displaystyle \\frac{\\var{p}z}{\\frac{\\var{p}}{\\var{p}z}}
&=\\var{p}z\\div{\\frac{\\var{p}}{\\var{p}z}}\\\\
&= \\var{p}z\\times{\\frac{\\var{p}z}{\\var{p}}}\\\\
&=\\frac{\\var{p}^2z^2}{\\var{p}}\\\\
&=\\var{p}z^2.\\end{align}$
Please fill in the gaps to simplify the fraction on the left.
\n| $\\simplify{{q}x^2y+{r}x*y+{s}x*y^2}$ | \n= | \n[[0]] | \n
| $\\var{t}xy$ | \n\n | [[1]] | \n
By looking for common factors, we see that $\\var{common}xy$ is a factor of every term (in the numerator and the denominator), no larger term is common, we call $\\var{common}xy$ the highest common factor. We divide the numerator and the denominator by the highest common factor to get the simplified fraction.
\nSometimes it is easier to factorise to find the common factor first and then cancel:
\n$\\begin{align}\\simplify{({q}x^2y+{r}x*y+{s}x*y^2)/({t}x*y)}&=\\dfrac{\\simplify{{common}x*y({list[0]}x+{list[1]}+{list[2]}y)}}{\\simplify{{common}x*y({list[3]})}}\\\\
&=\\dfrac{\\simplify{{list[0]}x+{list[1]}+{list[2]}y}}{\\simplify{{list[3]}}}.\\end{align}$
Students seem to freak out when their answer is not written exactly the same as the answer provided. This question tries to enforce that $(x-y)=-(y-x)$ and $\\frac{a-b}{c-d}=\\frac{b-a}{d-c}$
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "This question asks you to compared different looking answers, and determine if they are equivalent.
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", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "p", "q", "r", "s", "sym1", "sym2", "seed"], "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, "prompt": "Suppose you do a maths question and your answer is
\n\\[y=\\var{a*b}+\\frac{\\var{c}-x}{\\var{d}}.\\]
\nHowever, your friend has an answer of
\n\\[y=\\var{a*b} \\var{sym1}\\frac{x-\\var{c}}{\\var{d}}.\\]
\n\nThese answers are...
", "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": "Consider doing the subtraction $11-25$. Often people do the easier subtraction $25-11$, get $14$, and then they put a negative in front of it to conclude $11-25=-14$. This works because
\n\\[11-25=-(25-11)=-14.\\]
\nSo if we swap the order of subtraction, we need to put a negative out the front, but this is the same as just multiplying by $-1$ since $-(25-11)=-1\\times(25-11)$, which is also the same as dividing by $-1$.
\n\nTherefore, reversing the order of a subtraction is the same as multiplying (or dividing) by $-1$.
\n\na)
\nTo determine if $\\frac{\\var{c}-x}{\\var{d}}$ is equal to $\\frac{x-\\var{c}}{\\var{d}}$, notice the only difference is the subtraction in the numerator is reversed. But $\\var{c}-x\\ne x-\\var{c}$. So these answers are not the same!
\nTo determine if $\\frac{\\var{c}-x}{\\var{d}}$ is equal to $-\\frac{x-\\var{c}}{\\var{d}}$, notice
\n$\\begin{align}-\\frac{x-\\var{c}}{\\var{d}}&=\\frac{-(x-\\var{c})}{\\var{d}}\\\\&=\\frac{-x+\\var{c}}{\\var{d}}\\\\&=\\frac{\\var{c}-x}{\\var{d}}\\end{align}$
\nSo the negative out the front and the reversing of the subtraction cancelled each other out, and these answers are actually the same.
"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["equal!
", "not equal!
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\n\\[z=\\frac{\\var{a}x^\\var{p}-\\var{b}y^\\var{q}}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}.\\]
\nHowever, your friend has an answer of
\n\\[z=\\frac{\\var{b}y^\\var{q}-\\var{a}x^\\var{p}}{\\var{d}x^\\var{r}y^\\var{s}\\var{sym2}\\var{c}xy}.\\]
\n\nThese fractions are...
\n[[0]]
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\n\nYou should notice that these fractions are very similar except that the order of subtraction is reversed in the numerator and the denominator. We should know that reversing the order of subtraction introduces a negative out the front, if we do this twice we will have two negatives out the front, which of course means a positive! That is,
\n$\\begin{align}\\frac{\\var{a}x^\\var{p}-\\var{b}y^\\var{q}}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}&=\\frac{-(\\var{b}y^\\var{q}-\\var{a}x^\\var{p})}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}\\\\&=\\frac{-(\\var{b}y^\\var{q}-\\var{a}x^\\var{p})}{-(\\var{d}x^\\var{r}y^\\var{s}-\\var{c}xy)}\\\\&=\\frac{\\var{b}y^\\var{q}-\\var{a}x^\\var{p}}{\\var{d}x^\\var{r}y^\\var{s}-\\var{c}xy}\\end{align}$
\nSo the answers are the same!
\nYou should notice that in the numerator the order of subtraction has been swapped and in the denominator a $-\\var{d}x^\\var{r}y^\\var{s}$ has been replaced with $+\\var{d}x^\\var{r}y^\\var{s}$. These are not the same answers. If you require further proof, set them to be equal and see what happens, or even easier, substitute a value for $x$ and $y$ into both of them:
\nLet $x=1$ and $y=1$ and we will compare the fractions. For 'your' answer we get $\\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c-d)}}$ but for 'your friends' answer we get $\\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c+d)}}$ and therefore the fractions are not equal!
equivalent! Just multiply (or divide) the numerator and denominator by $-1$ to see this.
", "not equivalent! "], "matrix": ["if(seed=0,1,-1)", "if(seed=1,1,-1)"], "distractors": ["", ""]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Algebraic fractions: operations involving algebraic fractions", "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/"}], "variables": {"primes": {"definition": "shuffle([2,3,5,7,11,13,17])", "name": "primes", "description": "", "group": "numerical fractions", "templateType": "anything"}, "a": {"definition": "primes[0]", "name": "a", "description": "", "group": "numerical fractions", "templateType": "anything"}, "d": {"definition": "primes[3]", "name": "d", "description": "", "group": "numerical fractions", "templateType": "anything"}, "c": {"definition": "primes[2]", "name": "c", "description": "", "group": "numerical fractions", "templateType": "anything"}, "h": {"definition": "primes[6]", "name": "h", "description": "", "group": "numerical fractions", "templateType": "anything"}, "g": {"definition": "primes[5]", "name": "g", "description": "", "group": "numerical fractions", "templateType": "anything"}, "f": {"definition": "primes[4]", "name": "f", "description": "", "group": "numerical fractions", "templateType": "anything"}, "j": {"definition": "random(primes except [d,g,h])", "name": "j", "description": "", "group": "numerical fractions", "templateType": "anything"}, "b": {"definition": "primes[1]", "name": "b", "description": "", "group": "numerical fractions", "templateType": "anything"}}, "statement": "Evaluate the following and write your answer as a single fraction. Use / to signify a fraction or division, for example $\\frac{2a-1}{x+3}$ is written (2a-1)/(x+3). Simplify/cancel where possible.
", "parts": [{"showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"showFeedbackIcon": true, "type": "jme", "showPreview": true, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "checkingType": "absdiff", "vsetRangePoints": 5, "answer": "({a+1}x+{c})/{b}", "checkVariableNames": true, "answerSimplification": "simplifyFractions", "variableReplacements": [], "marks": 1, "customMarkingAlgorithm": "", "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "failureRate": 1, "unitTests": []}, {"showFeedbackIcon": true, "type": "jme", "showPreview": true, "valuegenerators": [{"name": "y", "value": ""}], "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "checkingType": "absdiff", "vsetRangePoints": 5, "answer": "{d-a}/({c}y)", "checkVariableNames": true, "answerSimplification": "simplifyFractions", "variableReplacements": [], "marks": 1, "customMarkingAlgorithm": "", "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "failureRate": 1, "unitTests": []}], "prompt": "$\\displaystyle\\frac{\\var{a}x}{\\var{b}}+\\frac{x+\\var{c}}{\\var{b}}=$ [[0]]
\n$\\displaystyle\\frac{\\var{d}}{\\var{c}y}-\\frac{\\var{a}}{\\var{c}y}=$ [[1]]
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\n\nThese fractions have a common denominator (the number on the bottom). This means they are out of the same number of parts and can be compared easily, for example, it is clear $\\frac{2}{3}$ is less than $\\frac{5}{3}$ but not so clear that $\\frac{3}{5}$ is less than $\\frac{2}{3}$.
\n\nLet's say you need to evaluate $\\frac{2}{3}+\\frac{5}{3}$, in words this is 'two thirds plus five thirds', so how many thirds are there in total? Seven thirds!
\nSo we have
\n\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]
\nThe same logic is used for subtraction. Suppose you had seven fourths and someone borrowed three fourths, then you are left with four fourths.
\nThat is
\n\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]
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\n$\\displaystyle\\simplify{(b+{h})/{f}-(b+{j})/{g}}=$ [[1]]
\n$\\displaystyle \\frac{\\var{a}}{\\var{d}r}+\\var{f}r=$ [[2]]
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\n\nIf your question was $\\frac{5}{4}+\\frac{3}{8}$ we could rewrite the first fraction as $\\frac{10}{8}$ (by multiplying the top and bottom by 2) and then both fractions would have a denominator of 8. At this point, we can perform the addition. Our working might look like this:
\n\\[\\frac{5}{4}+\\frac{3}{8}=\\frac{5\\times 2}{4\\times 2}+\\frac{3}{8}=\\frac{10}{8}+\\frac{3}{8}=\\frac{13}{8}\\]
\n\n\nOften we need to rewrite both fractions to get a common denominator, for instance, $\\frac{5}{4}-\\frac{2}{3}$. We could multiply the first fraction by 3 on the top and bottom, so that it's denominator was 12, and then multiply the second fraction by 4 on the top and bottom so that it also had a denominator of 12. Then we could perform the subtraction. Our working might look like this:
\n\\[\\frac{5}{4}-\\frac{2}{3}=\\frac{5\\times 3}{4\\times 3}-\\frac{2\\times 4}{3\\times 4}=\\frac{15}{12}-\\frac{8}{12}=\\frac{7}{12}\\]
\n\n\nAlso, recall that whole numbers are just fractions with a denominator of 1, for example $3=\\frac{3}{1}$.
\n\nIn general, the best denominator is the lowest common multiple (LCM) of the two denominators.
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\n$\\displaystyle -\\frac{\\var{f}+w}{\\var{j}}\\times \\var{d}=$ [[1]]
\n\n\n", "customName": "", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "scripts": {}, "steps": [{"showFeedbackIcon": true, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "Multiply the tops and the bottoms.
\n\nFor example
\n\\[\\frac{4}{5}\\times \\frac{2}{3}=\\frac{4\\times 2}{5 \\times 3}=\\frac{8}{15}\\]
\n\n\nAlso recall that whole numbers are just fractions with a denominator of 1, for example $7=\\frac{7}{1}$.
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\n$\\displaystyle \\frac{\\var{b}q}{\\var{c}q}\\div (\\var{d}+t)=$ [[1]]
\n$\\displaystyle \\var{j}z\\div \\left(\\frac{\\var{-d}(z+1)^2}{\\var{f}z}\\right)=$ [[2]]
\n", "customName": "", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "scripts": {}, "steps": [{"showFeedbackIcon": true, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "Flip the second fraction and then multiply.
\n\nFlipping a fraction is also known as taking the reciprocal of the fraction (or inverting a fraction). Note that a whole number is also a fraction with a denominator of 1, for example, $6=\\frac{6}{1}$.
\nHow do you find half of a number? You could 'divide it by 2', or you could 'multiply by $\\frac{1}{2}$. Notice that $\\frac{1}{2}$ is the reciprocal of 2. When we divide by a number this is actually the same as multiplying by its reciprocal.
\n\nSuppose you need to evaluate $\\frac{3}{7}\\div\\frac{5}{4}$. Recall this is the same as asking 'how many $\\frac{5}{4}$s are in $\\frac{3}{7}$?', but that doesn't seem to be very helpful here! What is helpful is realising that dividing by $\\frac{5}{4}$ is the same as multiplying by $\\frac{4}{5}$. Our working could look like this
\n\\[\\frac{3}{7}\\div\\frac{5}{4}=\\frac{3}{7}\\times\\frac{4}{5}=\\frac{3\\times 4}{7\\times 5}=\\frac{12}{35}\\]
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$\\displaystyle \\frac{\\frac{\\var{b}+x}{y+\\var{a}}}{\\frac{ \\var{b}}{\\var{a}}}=$ [[0]]
\n\n$\\displaystyle \\frac{\\frac{w+\\var{f}}{\\var{g}w}}{w+\\var{f}}=$ [[1]]
\n\n$\\displaystyle \\frac{\\var{j}r}{\\frac{\\var{h}r}{\\var{c}r}}=$ [[2]]
\n", "customName": "", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "scripts": {}, "steps": [{"showFeedbackIcon": true, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "The fraction bar means division.
\n\nThe fraction $\\frac{2}{3}$ means 2 divided by 3. So these questions are just division questions! It is important to note which fraction bar is big and which are small, so you know the order of the divisions.
\n\nHere are some examples:
\n\\[\\frac{7}{\\frac{5}{6}}=7\\div\\frac{5}{6} =7\\times\\frac{6}{5}=\\frac{42}{5}\\]
\n\\[\\frac{\\frac{7}{5}}{6}=\\frac{7}{5}\\div 6=\\frac{7}{5}\\times \\frac{1}{6}=\\frac{7}{30}\\]
\n\\[\\frac{\\frac{9}{11}}{\\frac{5}{3}}=\\frac{9}{11}\\div\\frac{5}{3}=\\frac{9}{11}\\times \\frac{3}{5}=\\frac{27}{55}\\]
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Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.
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