Question

Express the following recurring decimals as fractions.

$0.1\overline{3}$

Answer 1

Given, $0.1\overline{3}$

Let $x=0.1\overline{3}=0.13333.....\left(i\right)$

Multiplying both the sides by $10$ we get

$10x=1.333....\left(ii\right)$

Subtracting $\left(i\right)$ from $\left(ii\right)$ we get

$$\begin{gathered} 10x-x=1.333...-0.13333..... \\ \Rightarrow 9x=1.2 \\ \Rightarrow x=\frac{1.2}{9} \\ \Rightarrow x=\frac{12}{10}\times \frac{1}{9} \\ =\frac{12}{90}=\frac{2}{15} \end{gathered}$$

Hence, $0.1\overline{3}=\frac{2}{15}$