Question

Find the rational numbers having the following decimal expansions :

(i) $0.\overline{3}$

(ii) $0.\overline{231}$

(iii) $3.\overline{52}$

(iv) $0.6\overline{8}$

Answer 1

(i) $0.\overline{3}=0.3333$

$=0.3+0.03+0.003+...\infty$

$=\frac{3}{10}+\frac{3}{{10}^2}+\frac{3}{{10}^3}+....\infty$

$=\frac{3}{10}(1+\frac{1}{10}+\frac{1}{{10}^2}+.....)$

$=\frac{3}{10}\left(\frac{1}{1-\frac{1}{10}}\right)$ $[\because \text{Sum of G.P.}=\frac{a}{1-r}]$

$=\frac{3}{10}\times \frac{10}{9}$

$=\frac{3}{9}$

Hence, $0.\overline{3}=\frac{1}{3}$

(ii) $0.\overline{231}=0.231231231....$

$=0.231+0.000231+0.000000231+....\infty$

$=\frac{231}{{10}^3}=\frac{231}{{10}^6}+\frac{231}{{10}^9}+....\infty$

$=\frac{231}{{10}^3}(1+\frac{1}{{10}^3}+\frac{1}{{10}^6}+.....)$

$=\frac{231}{1000}\left(\frac{1}{1-\frac{1}{1000}}\right)$

Hence,$0.\overline{231}=\frac{231}{999}$

(iii) $3.\overline{52}=3+\mathrm{0.52222.....}$

$=3+0.5+0.02+0.002+0.0002+....\infty$

$=3.5+\frac{2}{{10}^2}+\frac{2}{{10}^3}+\frac{2}{{10}^4}+....\infty$

$=3.5+\frac{2}{{10}^2}(1+\frac{1}{10}+\frac{1}{{10}^2}+.....)$

$=\frac{35}{10}+\frac{2}{100}\left(\frac{1}{1-\frac{1}{10}}\right)$

$=\frac{35}{10}+\frac{2}{100}\times \frac{10}{9}$

$=\frac{35}{10}+\frac{2}{90}$

$=\frac{315+2}{90}$

Hence, $3.\overline{52}=\frac{317}{90}$

(iv) The rational number can be written as :

$0.6\overline{8}=0.6+0.08+0.008+0.0008+....\infty$

$=\frac{3}{5}+8[0.01+0.001+0.0001+....\infty ]$

$=\frac{3}{5}+8[\frac{1}{100}+\frac{1}{1000}+....\infty ]$

This is an infinite GP with first term $\frac{1}{100}$ and common ratio $\frac{1}{10}$

$=\frac{3}{5}+8.\frac{1}{100}.\frac{1}{1-\frac{1}{10}}$

$=\frac{3}{5}+\frac{4}{45}=\frac{31}{45}$

Hence, $0.\overline{52}=\frac{31}{45}$.