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Revision:Fractions

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Fractions

1 Fractions
2 A note on cancelling
- 2.1 Examples
- 2.2 Example
3 Comments

Fractions

Introduction

$\frac{1}{2}$ means 1 divided by 2. If you try this on a calculator, you will get an answer of 0.5 .

$\frac{3}{6}$ means 3 divided by 6. Using a calculator, you will find that this too gives an answer of 0.5 .

That is because $\frac{1}{2} = \frac{3}{6} = 0.5$ .

Fractions such as $\frac{3}{6}$ can be cancelled. You can divide the top and bottom of the fraction by 3 to get $\frac{1}{2}$ .

With fractions, you are allowed to multiply or divide the top and bottom of the fraction by some number, as long as you multiply (or divide) everything on the top and everything on the bottom by that number.

So:

$\displaystyle \frac{5}{12} = \frac{10}{24}$ (multiplying top and bottom by 2).

Adding and subtracting fractions

To add two fractions, the bottom (denominator) of the two fractions must be the same.

$\displaystyle \frac{1}{2} + \frac{3}{2} = \frac{4}{2}$ ;

$\displaystyle \frac{1}{10} + \frac{3}{10} + \frac{5}{10} = \frac{9}{10}$ .

If the denominators are not the same, multiply or divide the top and bottom of one (or both!) of the fractions by a whole number to make the denominator of each fraction the same.

Example

$\displaystyle \frac{5}{6} + \frac{2}{3} = \frac{5}{6} + \frac{2\times 2}{3\times 2} = \frac{5}{6} + \frac{4}{6} = \frac{5 + 4}{6} = \frac{9}{6} = \frac{3}{2}$ .

The same is true when subtracting fractions.

Multiplying fractions

This is simple, just multiply the two numerators (top bits) together, and the two denominators (bottom bits) together:

$\displaystyle \frac{2}{3} \times \frac{5}{8} = \frac{2\times 5}{3\times 8} = \frac{10}{24} = \frac{5}{12}$ .

Dividing Fractions

If A, B, C and D are any numbers,

$\displaystyle \frac{A}{B}\div \frac{C}{D} = \frac{A}{B}\times \frac{D}{C}$

So:

$\displaystyle \frac{1}{2}\div \frac{2}{3} = \frac{1}{2}\times \frac{3}{2} = \frac{3}{4}$ .

Harder examples

These rules work even when the fractions involve algebra.

$\displaystyle \frac{2x}{5}\div \frac{x}{3} = \frac{2x}{5} \times \frac{3}{x} = \frac{6x}{5x} = \frac{6}{5}$ (the x's cancel in the last step.)

A note on cancelling

Fractions, of course, can often be 'cancelled down' to make them simpler. For example, $\frac{4}{6} = \frac{2}{3}$ .

You can divide or multiply the top and bottom of any fraction by any number, as long as you do it to both the top and bottom. However, when there is more than one term on the top and/or bottom, to cancel you must divide every term in the top and bottom by that number.

Examples

$\displaystyle \frac{2 + x}{2}$ .

In this example, some people might try to cancel the 2s, but you cannot do this. You would have to divide the by 2 also, to get $1 + \frac{1}{2}x$ .

$\displaystyle \frac{2(x + 4)}{4} = \frac{x + 4}{2}$ .

Here there is only one term in the numerator (top) and denominator (bottom) of the fraction, so you can divide top and bottom by 2.

Example

Simplify the expression:

$\displaystyle \frac{2x^2 - 5x + 2}{x^2 - 4}$ .

In questions such as this, it is often useful to factorise first to get:

$\displaystyle \frac{(2x - 1)(x - 2)}{(x + 2)(x - 2)}$ .

Factorising means that there is now only one term in the numerator and denominator, whereas before there were two. We can now divide top and bottom by (x - 2) :