Question

Express the $3.\overline{3}$ rational numbers in the form $\frac{p}{q}$ , where $p$and $q$ are integers and $q\ne 0$.

Answer 1

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

Let , $x=3.\overline{3}=3.3333.....$ $--\left(1\right)$

Multiplying both sides by $10$, we get

$10x=33.333...=33.\overline{3}$ $--\left(2\right)$

If we subtract $3.\overline{3}$ from $33.\overline{3}$, the repeating portion of the decimal cancels out.
∴ Subtracting$\left(1\right)$ from $\left(2\right)$, we get

$$\begin{gathered} 10x-x=33.\overline{3}-3.\overline{3} \\ \Rightarrow 9x=30 \\ \Rightarrow x=\frac{30}{9} \\ \Rightarrow x=\frac{10}{3} \end{gathered}$$

Hence , the rational number $3.\overline{3}$ in the form $\frac{p}{q}$ , where $p$and $q$ are integers and $q\ne 0$ is $\frac{10}{3}$.