# How do you find the repeating decimal 0.45 with 45 repeated as a fraction?

Nov 6, 2016

$0. \overline{45} = \frac{5}{11}$

#### Explanation:

First, in case you have not encountered it before, note that you can indicate a repeating decimal by placing a bar over the repeating pattern:

$0. \overline{45} = 0.45454545 \ldots$

With that notation, see what happens when we multiply $0. \overline{45}$ by $\left(100 - 1\right)$:

$\left(100 - 1\right) 0. \overline{45} = 100 \cdot 0. \overline{45} - 1 \cdot 0. \overline{45}$

$\textcolor{w h i t e}{\left(100 - 1\right) 0. \overline{45}} = 45. \overline{45} - 0. \overline{45}$

$\textcolor{w h i t e}{\left(100 - 1\right) 0. \overline{45}} = 45$

The $100$ shifted our original decimal representation $2$ places to the left - the length of the repeating pattern.

Then subtracting the original cancelled out the repeating tail.

Next, divide both ends by $\left(100 - 1\right)$ and simplify to find:

$0. \overline{45} = \frac{45}{100 - 1} = \frac{45}{99} = \frac{5 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}}{11 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}} = \frac{5}{11}$