Question

Write the repeating decimal $0.\overline{8}$ as a fraction.

Short answer

Fraction form of$0.\overline{8}$is$\frac{8}{9}$.

Step-by-step solution

Step 1. Given information

Repeating decimal is$0.\overline{8}$.

Step 2. Write the repeating decimal as a fraction. 

Consider the repeating decimal $0.\overline{8}$

$\begin{array}{rcl}0.\overline{8}& =& 0.8888888...\\ & =& 0.8+0.08+0.008+0.0008+...\end{array}$

Let $a,r$denote the first term and the common ratio respectively.

$\begin{array}{rcl}a& =& 0.8\\ r& =& \frac{0.08}{0.8}\\ & =& 0.1\end{array}$

So,

$\begin{array}{rcl}{S}_n& =& \frac{a}{1-r}\\ 0.8+0.08+0.008+0.0008+...& =& \frac{0.8}{1-0.1}\\ & =& \frac{0.8}{0.9}\\ & =& \frac{8}{9}\end{array}$