Question

Express the $32.12\overline{35}$ 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 , $x=32.12\overline{35}=32.123535.....$ $--\left(1\right)$

Multiplying both sides by $100$, we get

$100x=3212.3535...=3212.\overline{35}$ $--\left(2\right)$

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

$10000x=321235.353535...=321235.\overline{35}$ $--\left(3\right)$

If we subtract $3212.\overline{35}$ from $321235.\overline{35}$, the repeating portion in the decimal cancels out.
∴ Subtracting$\left(2\right)$ from $\left(3\right)$, we get

$$\begin{gathered} 10000x-100x=321235.\overline{35}-3212.\overline{35} \\ \Rightarrow 9900x=318023 \\ \Rightarrow x=\frac{318023}{9900} \end{gathered}$$

Hence , the rational number $32.12\overline{35}$ in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$ is $\frac{318023}{9900}$.