Question

Find the $\frac{p}{q}$ form of the number $0.123\overline{456}$, where p and q are integers and $q\ne 0$.

Answer 1

Given number is $0.123\overline{456}$
Let, $x=0.123\overline{456}$
Multiply both sides of $x$ by 1000
$\Rightarrow 1000x=\mathrm{123.456456......}$ ---(1)
Now, multiply both sides of $x$ by 1000000
$\Rightarrow 1000000x=\mathrm{123456.456456.......}$ ---(2)

Subtracting (1) from (2), we get
$1000000x-1000x=\left(\mathrm{123456.456456...}\right)-\left(\mathrm{123.456456..}\right)$
$\Rightarrow$$999000x=123333$
$\Rightarrow x=\frac{123333}{999000}=\frac{41111}{333000}$