# Fractions

There are many rules and strategies for performing arithmetic with fractions. This section of the module will deal with those techniques required in computing.

We have already stated that a rational number can be written as a fraction (or ratio) of integers. The pattern for a rational number can be written as $\displaystyle{\frac{a}{b}}$ where $a$ and $b$ are integers and $b\ne 0$.

Note: the $\ne$ symbol means "not equal to".

The number on the top of the fraction is called the numerator, in this case $a$. The number on the bottom of the fraction is called the denominator, in this case $b$. $\displaystyle{\frac{\text{numerator}}{\text{denominator}}}$

### Equal fractions

Fractions of the same size are called equal (or equivalent) fractions.

Two fractions are equal if it is possible to multiply (or divide) the numerator and denominator of one fraction by the same number to obtain the other.

 1/4 1/4 1/4 1/4 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

$\displaystyle{\frac14}$ is equal to $\displaystyle{\frac28}$

Starting with $\displaystyle{\frac14}$, to convert this to a fraction in eighths, you need to multiply $4$ by $2$ to obtain $8$, so it is necessary to do the same to the numerator. $$\frac{1\times 2}{4\times 2}=\frac28$$

Starting with $\displaystyle{\frac28}$, to convert this to quarters, you need to divide $8$ by $2$ to obtain $4$, so it is necessary to do the same to the numerator. $$\frac{2\div 2}{8\div 2}=\frac14$$ This is an example of simplifying the fraction. The fraction is now in a ratio of smaller numbers.

Are the following pairs of fractions equal?

 $\displaystyle{\frac{8}{16},\;\frac12}$ Yes No $\displaystyle{\frac69,\;\frac13}$ Yes No $\displaystyle{\frac{25}{75},\;\frac27}$ Yes No $\displaystyle{\frac{75}{100},\;\frac34}$ Yes No

#### Simplifying Fractions

To simplify a fraction, divide both the numerator and the denominator by the same number.

##### Example

\begin{align*} \frac24 &=\frac{2\div{\color{red}2}}{4\div{\color{red}2}}\\[1ex] &=\frac12 \end{align*}

The only number that now divides into the numerator and denominator is $1$, so we consider $\displaystyle{\frac12}$ to be the simplest form of the fraction.

Sometimes you can use different methods to find the simplest form of a fraction but the final answer is always the same. Below are two different methods to find the simplest fraction equal to $\displaystyle{\frac{8}{16}}$.

 \begin{align*}\frac{8}{16} &=\frac{8\div{\color{red}2}}{16\div{\color{red}2}}\\[1ex] &=\frac{4\div{\color{red}2}}{8\div{\color{red}2}}\\[1ex] &=\frac{2\div{\color{red}2}}{4\div{\color{red}2}}\\[1ex]&=\frac12 \end{align*} \begin{align*} \frac{8}{16} &=\frac{8\div{\color{red}8}}{16\div{\color{red}8}}\\[1ex] &=\frac12 \end{align*}
Both methods are correct.
What is the best number to divide both numerator and denominator by?
Answer: the largest possible number that divides into both.
(The technical name for this number is "lowest common multiple".)

Are these fractions in the simplest form?

 $\displaystyle{\frac24}$ Yes No $\displaystyle{\frac{7}{10}}$ Yes No $\displaystyle{\frac{4}{82}}$ Yes No $\displaystyle{\frac{9}{27}}$ Yes No