Question

Express the $2.\overline{91}$ 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 , $x=2.\overline{91}=2.919191.....$ $--\left(1\right)$

Multiplying both sides by $100$, we get

$100x=291.919191...=291.\overline{91}$ $--\left(2\right)$

If we subtract $2.\overline{91}$ from $291.\overline{91}$, the repeating portion in the decimal cancels out.
∴ Subtracting$\left(1\right)$ from $\left(2\right)$, we get

$$\begin{gathered} 100x-x=291.\overline{91}-2.\overline{91} \\ \Rightarrow 99x=289 \\ \Rightarrow x=\frac{289}{99} \end{gathered}$$

Hence , the rational number $2.\overline{91}$ in the form $\frac{p}{q}$ , where $pandq$are integers and $q\ne 0$ is $\frac{289}{99}$.