Question

Express the $0.00\overline{32}$ rational numbers in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$.

Answer 1

Expressing rational number in $\frac{p}{q}$form , where $pandq$ are integers and $q\ne 0$ :-

Let , $x=0.00\overline{32}=0.00323232.....$ $--\left(1\right)$

Multiplying both sides by $100$, we get

$100x=0.3232...=0.\overline{32}$ $--\left(2\right)$

Multiplying equation$\left(1\right)$both sides by $10000$, we get

$10000x=32.3232...=32.\overline{32}$ $--\left(3\right)$

If we subtract $0.\overline{32}$ from $32.\overline{32}$, the repeating portion of the decimal cancels out.
∴ Subtracting$\left(2\right)$ from $\left(3\right)$, we get

$$\begin{gathered} 10000x-100x=32.\overline{32}-0.\overline{32} \\ \Rightarrow 9900x=32 \\ \Rightarrow x=\frac{32}{9900} \\ \Rightarrow x=\frac{8}{2475} \end{gathered}$$

Hence , the rational number $0.00\overline{32}$ in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$ is $\frac{8}{2475}$.