Question

Express the $2.\overline{317}$ 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 , $x=2.\overline{317}=2.317317317.....$ $--\left(1\right)$

Multiplying both sides by $1000$, we get

$1000x=2317.317317...=2317.\overline{317}$ $--\left(2\right)$

If we subtract $2.\overline{317}$ from $2317.\overline{317}$, the repeating portion in the decimal cancels out.
∴ Subtracting$\left(1\right)$ from $\left(2\right)$, we get

$$\begin{gathered} 1000x-x=2317.\overline{317}-2.\overline{317} \\ \Rightarrow 999x=2315 \\ \Rightarrow x=\frac{2315}{999} \end{gathered}$$

Hence , the rational number $2.\overline{317}$ in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$ is $\frac{2315}{999}$.