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Algebraic fractions involve various rules and some important tips are:

1.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.

2.When you have a fraction, say $\\displaystyle{\\frac{1+2a+3b}{5+x}}$, it represents the result of dividing everything on top by everything on the bottom, that is

\\[\\frac{1+2a+3b}{5+x}=\\frac{1}{5+x}+\\frac{2a}{5+x}+\\frac{3b}{5+x}\\]

3. Whenever 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.

Check this video for further guidance:

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

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

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Add the tops, leave the bottom the same.

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

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!

So we have

\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]

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

That is

\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]

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

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

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

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

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:

\\[\\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}\\]

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:

\\[\\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}\\]

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

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]]

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

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Multiply the tops and the bottoms.

For example

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

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

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

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

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

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Flip the second fraction and then multiply.

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

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.

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

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

$\\displaystyle \\frac{\\frac{w+\\var{f}}{\\var{g}w}}{w+\\var{f}}=$ [[1]]

$\\displaystyle \\frac{\\var{j}r}{\\frac{\\var{h}r}{\\var{c}r}}=$ [[2]]

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The fraction bar means division.

The 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.

Here are some examples:

\\[\\frac{7}{\\frac{5}{6}}=7\\div\\frac{5}{6} =7\\times\\frac{6}{5}=\\frac{42}{5}\\]

\\[\\frac{\\frac{7}{5}}{6}=\\frac{7}{5}\\div 6=\\frac{7}{5}\\times \\frac{1}{6}=\\frac{7}{30}\\]

\\[\\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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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.

NOTE: To write a fraction, type as - (Numerator)/(Denominator)

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Add, subtract, multiply and divide numerical fractions.

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