Question

Show that $\mathrm{0.2353535..}=0.2\overline{35}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

Answer 1

Let $x=\mathrm{0.2353535.......}$

$10x=\mathrm{2.353535.......}$ (1)

$1000x=\mathrm{235.3535.......}$ (2)

Subtract (1) from (2),

$1000x-10x=233.0000$

$990x=233$

$x=\frac{233}{990}=\frac{p}{q}$