Question

Express the $0.\overline{3}$ 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 , $x=0.\overline{3}=0.3333.....$ $--\left(1\right)$

Multiplying both sides by $10$, we get

$10x=3.333...=3.\overline{3}$ $--\left(2\right)$

If we subtract $0.\overline{3}$ from $3.\overline{3}$, the repeating portion in the decimal cancels out.
∴ Subtracting$\left(1\right)$ from $\left(2\right)$, we get

$$\begin{gathered} 10x-x=3.\overline{3}-0.\overline{3} \\ \Rightarrow 9x=3 \\ \Rightarrow x=\frac{3}{9} \\ \Rightarrow x=\frac{1}{3} \end{gathered}$$

Hence , the rational number $0.\overline{3}$ in the form of $\frac{p}{q}$ will be $\frac{1}{3}$, where p and q are integers and $q\ne 0$ .