Question:
The value of $0 . \overline{2}$ in the form $\frac{p}{g}$, where $p$ and $q$ are integers and $q \neq 0$, is
(a) $\frac{1}{5}$
(b) $\frac{2}{9}$
(c) $\frac{2}{5}$
(d) $\frac{1}{8}$
Solution:
Let $x=0 . \overline{2}=0.222 \ldots \quad \ldots$ (1)
Multiplying both sides by 10, we get
$10 x=2 . \overline{2} \quad \ldots \ldots(2)$
Subtracting (1) from (2), we get
$10 x-x=2 . \overline{2}-0 . \overline{2}$
$\Rightarrow 9 x=2$
$\Rightarrow x=\frac{2}{9}$
$\therefore 0 . \overline{2}=\frac{2}{9}$
Hence, the correct answer is option (b).