Question

Show that $1.\overline{34}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne o$.

Answer 1

Let the given number be,

$x=1.\overline{34}$

$x=\mathrm{1.34343434...}.....\left(i\right)$

On multiplying $100$ on both sides, we get:

$100x=\mathrm{134.343434...}....\left(ii\right)$

Subtracting $\left(i\right)$ from $\left(ii\right)$, we get:

$100x-x=\mathrm{134.343434...}-\mathrm{1.343434...}=133$

$\Rightarrow 99x=133$

$\therefore x=\frac{133}{99}$