Find the rational numbers having the following decimal expansions:
(i) $0 . \overline{3}$
(ii) $0 . \overline{231}$
(iii) $3.5 \overline{2}$
(iv) $0.6 \overline{8}$
(i) $0 . \overline{3}$
Let $S=0 . \overline{3}$
$\Rightarrow \mathrm{S}=0.3+0.03+0.003+0.0003+0.00003+\ldots \infty$
$\Rightarrow \mathrm{S}=0.3\left(1+10^{-1}+10^{-2}+10^{-3}+10^{-4}+\ldots \infty\right)$
$\mathrm{S}$ is a geometric series with the first term, $a$, being 1 and the common ratio, $r$, being $10^{-1} .$
$\therefore \mathrm{S}=\frac{1}{1-r}$
$\Rightarrow \mathrm{S}=0.3\left(\frac{1}{1-10^{-1}}\right)$
$\Rightarrow \mathrm{S}=\frac{3}{9}=\frac{1}{3}$
(ii) $0 . \overline{231}$
Let $S=0 . \overline{231}$
$\Rightarrow \mathrm{S}=0.231+0.000231+0.000000231+\ldots \infty$
$\Rightarrow \mathrm{S}=0.231\left(1+10^{-3}+10^{-6}+\ldots \infty\right)$
It is a G.P.
$\therefore \mathrm{S}=0.231\left(\frac{1}{1-10^{-3}}\right)$
$\Rightarrow \mathrm{S}=\frac{231}{999}$
(iii) $3.5 \overline{2}$
Let $\mathrm{S}=3.5 \overline{\overline{2}}$
$\Rightarrow \mathrm{S}=3.5+0.02+0.002+0002+0.00002+\ldots \infty$
$\Rightarrow \mathrm{S}=3.5+0.02\left(1+10^{-1}+10^{-2}+10^{-3}+10^{-4}+\ldots \infty\right)$
It is a G.P.
$\therefore \mathrm{S}=3.5+0.02\left(\frac{1}{1-10^{-1}}\right)$
$\Rightarrow \mathrm{S}=3.5+\frac{0.2}{9}$
$\Rightarrow \mathrm{S}=\frac{317}{90}$
(iv) $0.6 \overline{8}$
Let $S=0.6 \overline{8}$
$\Rightarrow \mathrm{S}=0.6+0.08+0.008+0.0008+0.00008+\ldots \infty$
$\Rightarrow \mathrm{S}=0.6+0.08\left(1+10^{-1}+10^{-2}+10^{-3}+\ldots \infty\right)$
It is a G.P.
$\therefore \mathrm{S}=0.6+0.08\left(\frac{1}{1-10^{-1}}\right)$
$\Rightarrow \mathrm{S}=0.6+\frac{0.8}{9}$
$\Rightarrow \mathrm{S}=\frac{6.2}{9}$
$\Rightarrow \mathrm{S}=\frac{62}{90}=\frac{31}{45}$
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