Collecting like terms, expanding a bracket (distributive law), substitution of values, finding factors, simplifying algebraic fractions.
"}, "duration": 0, "navigation": {"allowregen": true, "showresultspage": "oncompletion", "startpassword": "", "preventleave": true, "onleave": {"action": "none", "message": ""}, "reverse": true, "browse": true, "showfrontpage": true}, "question_groups": [{"name": "Group", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Collecting like terms", "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/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Simplify the following by collecting like terms.
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", "stepsPenalty": "1", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "steps": [{"customName": "", "customMarkingAlgorithm": "", "scripts": {}, "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).
\n\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 $2a+3a=5a$.
\n\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\\[\\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}\\]
\nWe present this as the sum of three unlike terms:
\n\\[\\simplify{{sum(a)}x+{sum(b)}y+{sum(c)}z}\\]
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\n\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 $2x^2+3x^2=5x^2$.
\n\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\\[\\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]})}\\]
\nWe present this as the sum of three unlike terms:
\n\\[\\simplify{{sum(d)}x^2+{sum(f)}x+{sum(g)}}\\]
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\n$\\begin{align*}\\simplify{{nmult}({nxcoeff}a+{nconstant})}&=\\simplify[expandBrackets]{{nmult}({nxcoeff}a+{nconstant})}\\\\&=\\simplify[!noLeadingMinus]{{nmult*nxcoeff}a+{nmult*nconstant}}\\end{align*}$
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\n[[0]]$a$ + [[1]]
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\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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Which of the following are factors of $\\simplify{{prime^primepower}x^{xpower}y^{ypower}(x+{xconstant})^{xconstantpower} (z+{failconstant})^{failsafepower}}$?
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\n\nWhat has {Name[0][0]} done to the first fraction in order to get the second? {Name[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": "If we multiply or divide the top and bottom of a fraction by a number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.
\n\nRecall $\\frac{3}{6}$ is equivalent to $\\frac{1}{2}$, notice we can divide the top and bottom of $\\frac{3}{6}$ by 3 to get $\\frac{1}{2}$. Similarly, if we divide the top and bottom of $\\frac{10x-50y}{20a+10}$ by $10$ we would get the equivalent fraction $\\frac{x-5y}{2a+1}$.
\n\nWhen simplifying fractions we try to get the fraction so there is no common factor on the top and the bottom (if there was we would divide by it). It is important that this common factor is common to all the terms and not just a couple, otherwise dividing by it would leave you with a fraction on a fraction.
"}], "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": "{Name[1][0]} has written $\\displaystyle{\\simplify{((x+{ar})/({f}x))/({br}y+{dr})}}$ in the equivalent form $\\displaystyle{\\simplify{(x+{ar})/({f*br}x*y+{f*dr}x)}}$.
\n\nWhat has {Name[1][0]} done to the first fraction in order to get the second? {Name[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": "If we multiply or divide the top and bottom of a fraction by a number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.
\n\nRecall $\\frac{1}{2}$ is equivalent to $\\frac{3}{6}$, notice we can multiply the top and bottom of $\\frac{1}{2}$ by 3 to get $\\frac{3}{6}$. Similarly, we can multiply the top and bottom of $\\frac{\\frac{x-3}{2x}}{x+1}$ by $2x$ to get the equivalent fraction $\\frac{x-3}{2x^2+2x}$.
\n\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 by the denominator of the smaller/inner 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, say $\\displaystyle{\\frac{1+2a+3b}{5}}$, it represents the result of dividing everything on top by everything on the bottom, that is
\n\n\\[\\frac{1+2a+3b}{5+x}=\\frac{1}{5+x}+\\frac{2a}{5+x}+\\frac{3b}{5+x}\\]
"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "choices": ["$\\var{g}$
", "$\\simplify{{g-1}x+{h}}$
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"], "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, say $\\displaystyle{\\frac{1+2a+3b}{5}}$, it represents the result of dividing everything on top by everything on the bottom, that is
\n\n\\[\\frac{1+2a+3b}{5+x}=\\frac{1}{5+x}+\\frac{2a}{5+x}+\\frac{3b}{5+x}\\]
"}], "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.
\n\nCompare the two denominators, what has happened? We have divided the first denominator by $(x+\\var{m})$ to get the second denominator. The same must be done to the numerator, but something divided by itself is $1$.
"}], "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": "1", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["/"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "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": "The expression $\\displaystyle{\\frac{\\var{p}z}{\\frac{\\var{p}}{\\var{p}z}}}$ can be simplified to [[0]] .
\nIf we multiply or divide the top and bottom of a fraction by a number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.
\n\nRecall $\\frac{1}{2}$ is equivalent to $\\frac{3}{6}$, notice we can multiply the top and bottom of $\\frac{1}{2}$ by 3 to get $\\frac{3}{6}$. Similarly, we can multiply the top and bottom of $\\frac{\\frac{x-3}{2x}}{x+1}$ by $2x$ to get the equivalent fraction $\\frac{x-3}{2x^2+2x}$.
\n\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 by the denominator of the smaller/inner 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": "{p}z^2", "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": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "z", "value": ""}]}], "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": "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), there is also no larger term that 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.
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", "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.
", "advice": "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.
\n\nb)
\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!
1
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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...
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", "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...
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", "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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