Question

Find the rational numbers having the following decimal expansions:
(i) $0.\overline{3}$
(ii) $0.\overline{231}$
(iii) $3.5\overline{2}$
(iv) $0.6\overline{8}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)0.\overline{3} \\ \mathrm{Let}\mathrm{S}=0.\overline{3} \\ \Rightarrow \mathrm{S}=0.3+0.03+0.003+0.0003+0.00003+...\infty \\ \Rightarrow \mathrm{S}=0.3\left(1+{10}^{-1}+{10}^{-2}+{10}^{-3}+{10}^{-4}+...\infty \right) \\ \mathrm{S}\mathrm{is}\mathrm{a}\mathrm{geometric}\mathrm{series}\mathrm{with}\mathrm{the}\mathrm{first}\mathrm{term},a,\mathrm{being}1\mathrm{and}\mathrm{the}\mathrm{common}\mathrm{ratio},r,\mathrm{being}{10}^{-1}. \\ \therefore \mathrm{S}=\frac{1}{1-r} \\ \Rightarrow \mathrm{S}=0.3\left(\frac{1}{1-{10}^{-1}}\right) \\ \Rightarrow \mathrm{S}=\frac{3}{9}=\frac{1}{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)0.\overline{231} \\ \mathrm{Let}\mathrm{S}=0.\overline{231} \\ \Rightarrow \mathrm{S}=0.231+0.000231+0.000000231+...\infty \\ \Rightarrow \mathrm{S}=0.231\left(1+{10}^{-3}+{10}^{-6}+...\infty \right) \\ \mathrm{It}\mathrm{is}\mathrm{a}\mathrm{G}.\mathrm{P}. \\ \therefore \mathrm{S}=0.231\left(\frac{1}{1-{10}^{-3}}\right) \\ \Rightarrow \mathrm{S}=\frac{231}{999} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)3.5\overline{2} \\ \mathrm{Let}\mathrm{S}=3.5\overline{2} \\ \Rightarrow \mathrm{S}=3.5+0.02+0.002+0002+0.00002+...\infty \\ \Rightarrow \mathrm{S}=3.5+0.02\left(1+{10}^{-1}+{10}^{-2}+{10}^{-3}+{10}^{-4}+...\infty \right) \\ \mathrm{It}\mathrm{is}\mathrm{a}\mathrm{G}.\mathrm{P}. \\ \therefore \mathrm{S}=3.5+0.02\left(\frac{1}{1-{10}^{-1}}\right) \\ \Rightarrow \mathrm{S}=3.5+\frac{0.2}{9} \\ \Rightarrow \mathrm{S}=\frac{317}{90} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)0.6\overline{8} \\ \mathrm{Let}\mathrm{S}=0.6\overline{8} \\ \Rightarrow \mathrm{S}=0.6+0.08+0.008+0.0008+0.00008+...\infty \\ \Rightarrow \mathrm{S}=0.6+0.08\left(1+{10}^{-1}+{10}^{-2}+{10}^{-3}+...\infty \right) \\ \mathrm{It}\mathrm{is}\mathrm{a}\mathrm{G}.\mathrm{P}. \\ \therefore \mathrm{S}=0.6+0.08\left(\frac{1}{1-{10}^{-1}}\right) \\ \Rightarrow \mathrm{S}=0.6+\frac{0.8}{9} \\ \Rightarrow \mathrm{S}=\frac{6.2}{9} \\ \Rightarrow \mathrm{S}=\frac{62}{90}=\frac{31}{45} \end{gathered}$$