Question

Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.
$0.\overline{6}$

Answer 1

Let $x=0.\overline{6}$ (Pure recurring decimal)
i.e. $x=0.6666\dots$(1)
Multiplying (1) by $10$, because ones digit i.e. $6$ is repeating, we get
$10x=6.6666$ …(2)
Subtracting (1) from (2), we get
$10x-x=(6.6666\dots )–(0.6666\dots )$
$\Rightarrow 9x=6$
$\Rightarrow x=\frac{6}{9}$
$\Rightarrow x=\frac{2}{3}$

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