Question

The simplest form of $1.\overline{6}$ is _______.

Answer 1

$$\begin{gathered} \mathrm{Let}x=1.6666........\left(1\right) \\ \mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 10x=16.6666........\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 10x-x=16.6666....-1.6666.... \\ \Rightarrow 9x=15 \\ \Rightarrow x=\frac{15}{9} \\ \Rightarrow x=\frac{5}{3} \end{gathered}$$

Hence, the simplest form of $1.\overline{6}$ is $\underline{\frac{5}{3}}$.