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Add, subtract, multiply and divide algebraic fractions.

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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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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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$\\displaystyle\\frac{\\var{a}x}{\\var{b}}+\\frac{x+\\var{c}}{\\var{b}}=$ [[0]]

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$\\displaystyle\\frac{\\var{d}}{\\var{c}y}-\\frac{\\var{a}}{\\var{c}y}=$ [[1]]

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Add the tops, leave the bottom the same.

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These 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 $\\frac{2}{3}$.

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Let'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!

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So we have

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\$\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\$

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The same logic is used for subtraction. Suppose you had seven fourths and someone borrowed three fourths, then you are left with four fourths.

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That is

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\$\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\$

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$\\displaystyle\\simplify{(a+{f})/{g}+({h}a+1)/{j}}=$ [[0]]

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$\\displaystyle\\simplify{(b+{h})/{f}-(b+{j})/{g}}=$ [[1]]

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$\\displaystyle \\frac{\\var{a}}{\\var{d}r}+\\var{f}r=$ [[2]]

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Rewrite the fractions so they have a common denominator. Then perform the addition or subtraction as required.

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If 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:

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\$\\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}\$

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Often 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:

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\$\\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}\$

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Also, recall that whole numbers are just fractions with a denominator of 1, for example $3=\\frac{3}{1}$.

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In general, the best denominator is the lowest common multiple (LCM) of the two denominators.

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$\\displaystyle\\frac{m+1}{n+1}\\times \\frac{y}{x}=$ [[0]]

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$\\displaystyle -\\frac{\\var{f}+w}{\\var{j}}\\times \\var{d}=$ [[1]]

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Multiply the tops and the bottoms.

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For example

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\$\\frac{4}{5}\\times \\frac{2}{3}=\\frac{4\\times 2}{5 \\times 3}=\\frac{8}{15}\$

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Also recall that whole numbers are just fractions with a denominator of 1, for example $7=\\frac{7}{1}$.

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$\\displaystyle{\\simplify{({f}+{a}x)^2/{h}}}\\div \\simplify{(({f}+{a}x){g})/({j}x)}=$ [[0]]

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$\\displaystyle \\frac{\\var{b}q}{\\var{c}q}\\div (\\var{d}+t)=$ [[1]]

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$\\displaystyle \\var{j}z\\div \\left(\\frac{\\var{-d}(z+1)^2}{\\var{f}z}\\right)=$ [[2]]

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Flip the second fraction and then multiply.

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Flipping 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, $6=\\frac{6}{1}$.

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How 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.

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Suppose 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

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\$\\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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