Question

Express the following recurring decimals as fractions.

$0.\overline{6}$

Answer 1

Expressing $0.\overline{6}$ as fractions:

Given, $0.\overline{6}$

Let $x=0.\overline{6}=0.666.....\left(i\right)$

Multiplying both the sides by $10$ we get

$10x=6.666....\left(ii\right)$

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

$$\begin{gathered} 10x-x=6.666.....-0.666... \\ \Rightarrow 9x=6 \\ \Rightarrow x=\frac{6}{9}=\frac{2}{3} \end{gathered}$$

Hence, $0.\overline{6}=\frac{2}{3}$.