Question

Find the rational number whose decimal expansion is $0.4\overline{23}$.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{rational}\mathrm{number}\mathrm{S}\mathrm{be}0.4\overline{23}. \\ \because \mathrm{S}=0.4\overline{23}=0.4+0.023+0.00023+0.0000023+...\infty \\ \Rightarrow \mathrm{S}=0.4+0.023\left[1+{10}^{-2}+{10}^{-4}+...\infty \right] \\ \mathrm{Clearly},\mathrm{S}\mathrm{is}\mathrm{a}\mathrm{geometric}\mathrm{series}\mathrm{with}\text{the}\mathrm{first}\mathrm{term},a,\text{being}1\mathrm{and}\text{the}\mathrm{common}\mathrm{ratio},\mathit{}r,\mathrm{being}{10}^{-2}. \\ \therefore \mathrm{S}=0.4+0.023\left[\frac{1}{1-{10}^{-2}}\right] \\ \Rightarrow \mathrm{S}=0.4+\frac{2.3}{99} \\ \Rightarrow \mathrm{S}=\frac{419}{990} \end{gathered}$$