Question

The value of $0.\overline{2}$ in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0, is
(a) $\frac{1}{5}$

(b) $\frac{2}{9}$

(c) $\frac{2}{5}$

(d) $\frac{1}{8}$

Answer 1

Let $x=0.\overline{2}=0.222........\left(1\right)$
Multiplying both sides by 10, we get
$10x=2.\overline{2}.....\left(2\right)$
Subtracting (1) from (2), we get
$$\begin{gathered} 10x-x=2.\overline{2}-0.\overline{2} \\ \Rightarrow 9x=2 \\ \Rightarrow x=\frac{2}{9} \\ \therefore 0.\overline{2}=\frac{2}{9} \end{gathered}$$
Hence, the correct answer is option (b).