Question

Show that $\mathrm{0.2353535.....}$ can be expressed in the form of $\frac{p}{q},$ where $p$ and $q$ are integers and $q\ne 0.$

Answer 1

Let $k=0.2353535.....$ ___(1)

$10k=\mathrm{2.353535...}$

$1000k=235.353535...$ ___(2)

$\left(2\right)-\left(1\right)$

$\Rightarrow (1000k-10k)=235.3535..-\mathrm{0.3535...}$

$=235$

$\Rightarrow 999k=235$

$\Rightarrow k=\frac{235}{999}$

Hence $\mathrm{0.2353535...}=\frac{235}{999}=\frac{p}{q}$