Question

Express $0.\overline{23}$ as a rational number in simplest form.

Answer 1

​Let x be $0.\overline{23}$.

$x=0.\overline{23}$ .....(1)
Multiplying both sides by 100, we get
$100x=23.\overline{23}$ .....(2)

Subtracting (1) from (2), we get
$$\begin{gathered} 100x-x=23.\overline{23}-0.\overline{23} \\ \Rightarrow 99x=23 \\ \Rightarrow x=\frac{23}{99} \end{gathered}$$

Thus, simplest form of $0.\overline{23}$ as a rational number is $\frac{23}{99}$.