Question

Show that $\mathrm{1.272727....}=1.\overline{27}$ can be expressed in the form $\frac{p}{q}$.

Answer 1

Let $x=1.\overline{27}..................\left(1\right)$

Then, $\left(1\right)\times 100\Rightarrow$ $100x=127.\overline{27}............\left(2\right)$

Subtracting (1) from (2),

$99x=126\Rightarrow x=\frac{126}{99}$.

Therefore, $\mathrm{1.272727...}=1.\overline{27}$ can be expressed in $\frac{p}{q}$, where $p=126$ and $q=99$.

$\therefore$ Hence, showed.