Add the tops, leave the bottom the same.
\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\\]
\n"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "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]]
"}, {"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "gapfill", "customName": "", "stepsPenalty": "2", "sortAnswers": false, "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "a", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "({j+g*h}a+{f*j+g})/{g*j}", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "b", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "({g-f}b+{g*h-f*j})/{f*g}", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "r", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "({a}+{f*d}r^2)/({d}r)", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}], "steps": [{"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "information", "customName": "", "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Rewrite the fractions so they have a common denominator. Then perform the addition or subtraction as required.
\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.
"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\displaystyle\\simplify{(a+{f})/{g}+({h}a+1)/{j}}=$ [[0]]
\n$\\displaystyle\\simplify{(b+{h})/{f}-(b+{j})/{g}}=$ [[1]]
\n$\\displaystyle \\frac{\\var{a}}{\\var{d}r}+\\var{f}r=$ [[2]]
"}, {"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "gapfill", "customName": "", "stepsPenalty": "2", "sortAnswers": false, "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "m", "value": ""}, {"name": "n", "value": ""}, {"name": "x", "value": ""}, {"name": "y", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": 1, "answer": "((m+1)*y)/((n+1)*x)", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "w", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": 1, "answer": "-({d*f}+{d}w)/{j}", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}], "steps": [{"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "information", "customName": "", "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "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 $7=\\frac{7}{1}$.
"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\displaystyle\\frac{m+1}{n+1}\\times \\frac{y}{x}=$ [[0]]
\n$\\displaystyle -\\frac{\\var{f}+w}{\\var{j}}\\times \\var{d}=$ [[1]]
\n\n\n"}, {"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "gapfill", "customName": "", "stepsPenalty": "2", "sortAnswers": false, "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "x", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "({a*j}*x^2+{j*f}*x)/{g*h}", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "t", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "{b}/({c}({d}+t))", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": false, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "z", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "({j*f}z^2)/({-d}(z+1)^2)", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}], "steps": [{"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "information", "customName": "", "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "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}\\]
\n"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "
$\\displaystyle{\\simplify{({f}+{a}x)^2/{h}}}\\div \\simplify{(({f}+{a}x){g})/({j}x)}=$ [[0]]
\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"}, {"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "gapfill", "customName": "", "stepsPenalty": "2", "sortAnswers": false, "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}, {"name": "c", "value": ""}, {"name": "d", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "(d*({b}a+b))/({b}a*(c+d))", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "w", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "1/({g}w)", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}, {"checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "useCustomName": false, "checkVariableNames": true, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "r", "value": ""}], "checkingType": "absdiff", "answerSimplification": "simplifyFractions", "scripts": {}, "vsetRangePoints": 5, "type": "jme", "vsetRange": [0, 1], "marks": "1", "answer": "{j*c}r/{h}", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customName": "", "unitTests": [], "showCorrectAnswer": true, "showPreview": true}], "steps": [{"scripts": {}, "adaptiveMarkingPenalty": 0, "type": "information", "customName": "", "useCustomName": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "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}\\]
"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\displaystyle \\frac{\\frac{\\var{b}a+b}{c+d}}{\\frac{ \\var{b}a}{d}}=$ [[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"}], "ungrouped_variables": [], "tags": [], "advice": "Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.
", "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "Add, subtract, multiply and divide
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.
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