Question

How do you convert $0.\overline{9}$ ($9$ being repeated) to a fraction?

Answer 1

Convert the decimal to a fraction.

Let, $x=0.9\overline{9}...\left(\mathrm{i}\right)$

Multiply both sides by $10$.

$10x=9.9\overline{9}...\left(\mathrm{ii}\right)$

Subtract $\left(\mathrm{i}\right)$ from $\left(\mathrm{ii}\right)$:

$$\begin{gathered} 10x-x=9.9\overline{9}-0.9\overline{9} \\ \Rightarrow 9x=9 \\ \Rightarrow x=\frac{1}{1} \end{gathered}$$

Therefore, the fraction form is $\frac{\mathbf{1}}{\mathbf{1}}$.