Question

Express as a fraction in simplest form $0.3\overline{65}$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{x}=0.3656565..............(\mathrm{equation}1) \\ \Rightarrow 10\mathrm{x}=3.656565..............\left(\mathrm{equation}2\right)\left(\mathrm{multiply}\mathrm{by}10\mathrm{in}\mathrm{equation}1\right) \\ \Rightarrow 100\mathrm{x}=36.565656................(\mathrm{equation}3)(\mathrm{multiply}\mathrm{by}10\mathrm{in}\mathrm{equation}2) \\ \Rightarrow 1000\mathrm{x}=365.656565..................(\mathrm{equation}4)(\mathrm{multiply}\mathrm{by}10\mathrm{in}\mathrm{equation}3) \\ \mathrm{Substracting}\mathrm{equation}\left(2\right)\mathrm{from}\mathrm{equation}\left(4\right), \\ \Rightarrow 1000\mathrm{x}-10\mathrm{x}=365.656565....-3.656565.... \\ \Rightarrow 990\mathrm{x}=362 \\ \Rightarrow \mathrm{x}=\frac{362}{990}\left(\mathrm{Divide}\mathrm{by}2\mathrm{in}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\right) \\ \Rightarrow \mathrm{x}=\frac{\mathbf{181}}{\mathbf{495}} \end{gathered}$$

Final answer:

Hence, the required fraction form of $0.3\overline{65}$$\mathbf{=}\frac{\mathbf{181}}{\mathbf{495}}$