Question

Write $0.\overline{32}$ in fraction form.

Answer 1

Write $0.\overline{32}$ in fraction form.

Let,

$x=0.\overline{32}...\left(1\right)$

The last two digits are recurring.

Multiply by $100$ both sides.

$$\begin{gathered} 100x=100\times 0.\overline{32} \\ \Rightarrow 100x=32.\overline{32}...\left(2\right) \end{gathered}$$

Subtract equation $\left(1\right)$ from equation $\left(2\right)$.

$\begin{array}{rcl}100x-x& =& 0.\overline{32}-32.\overline{32}\\ & \Rightarrow & 99x=32\\ \therefore x& =& \frac{32}{99}\end{array}$

Thus, $0.\overline{32}$ in fraction form is $\frac{32}{99}$.