Question

Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

$0.\overline{6}$

Answer 1

Express in terms of $\frac{p}{q}$:

$0.\overline{6}$

The given rational number is $0.\overline{6}=0.666...$

Assume that $x=0.666\dots$ $....\left(1\right)$

Then, $10x=6.666\dots$ $....\left(2\right)$

So, on subtracting $\left(1\right)$ from $\left(2\right)$

$10x–x=(6.6666\dots ..)–(0.66666\dots \dots ..)$

$\Rightarrow$. $9x=6$

$\Rightarrow$. $x=\frac{6}{9}$

$\Rightarrow$. $x=\frac{2}{3}$

Hence, $x$ is in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

Therefore, the given rational number can be expressed as $\frac{2}{3}$.