Question

Express the following to the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

$0.\overline{6}$

Answer 1

Given decimal is $0.\overline{6}$

Let $x=0.\overline{6}$

$\Rightarrow x=\mathrm{0.6666.....}$........ $\left(1\right)$

Multiplying equation $\left(1\right)$ with $10$ [Since, the number of digits in recurring part (Periodicity) is $1$]

$\Rightarrow 10x=10\times \left(\mathrm{0.666...}\right)$

$\Rightarrow 10x=\mathrm{6.666.....}$ .......... $\left(2\right)$

Lets do $\left(2\right)-\left(1\right)$

$10x-x=\left(\mathrm{6.6666...}\right)-\left(\mathrm{0.6666....}\right)$

$\Rightarrow 9x=\mathrm{6.000...}$

$\Rightarrow 9x=6$

$\Rightarrow x=\frac{6}{9}$

$\Rightarrow x=\frac{2}{3}$

Therefore, the $\frac{p}{q}$ form of $0.\overline{6}$ is $\frac{2}{3}$.