Question

Write the repeating decimal $0.\overline{4}$as a fraction.

Short answer

Fraction form of $0.\overline{4}$is equal to$\frac{4}{9}$.

Step-by-step solution

Step 1. Given information

Repeating decimal is$0.\overline{4}$.

Step 2. Write the repeating decimal as a fraction. 

The repeating decimal is $0.\overline{4}$.

$\begin{array}{rcl}0.\overline{4}& =& 0.444444....\\ & =& 0.4+0.04+0.004+0.0004+...\end{array}$

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

$\begin{array}{rcl}a& =& 0.4\\ r& =& \frac{0.04}{0.4}\\ & =& 0.1\end{array}$

$\begin{array}{rcl}{S}_n& =& \frac{a}{1-r}\\ & =& \frac{0.4}{1-0.1}\\ & =& \frac{4}{9}\end{array}$