# How do you write the repeating decimal 0.5 as a fraction?

##### 1 Answer
Oct 9, 2017

$0.55555 \ldots \ldots . = 0. \overline{5} = \frac{5}{9}$

#### Explanation:

There are two ways.

First consider $0.55555 \ldots \ldots .$ as an infinite geometric series

$\frac{5}{10} + \frac{5}{10} ^ 2 + \frac{5}{10} ^ 3 + \frac{5}{10} ^ 4 + \frac{5}{10} ^ 5 + \ldots \ldots \ldots \ldots \ldots \ldots$

with first term $a + 1 = \frac{5}{10}$ and common ratio $r = \frac{1}{10}$

Hence sum of the series is $\frac{a}{1 - r} = \frac{\frac{5}{10}}{1 - \frac{1}{10}} = \frac{\frac{5}{10}}{\frac{9}{10}} = \frac{5}{9}$

Second and easier is to consider $x = 0.55555 \ldots \ldots .$

and hence $10 x = 5.55555 \ldots \ldots \ldots \ldots \ldots \ldots$

and subtracting former from latter

$9 x = 5$ i.e. $x = \frac{5}{9}$