Question

When the repeating decimal $0.363636$..... is written in simplest fractional from $\frac{p}{q}$, find the sum $p+q$.

Answer 1

Let $x=\mathrm{0.363636...}$ ....$\left(1\right)$

$\Rightarrow 100\times x=100\times \mathrm{0.363636...}$

$\Rightarrow 100x=\mathrm{36.363636...}$ ....$\left(2\right)$

$\Rightarrow \left(2\right)-\left(1\right)\Rightarrow (100x-x)=\mathrm{36.363636...}-0.363636$

$\Rightarrow 99x=36$

$\Rightarrow x=\frac{36}{99}$

$\Rightarrow x=\frac{4}{11}$.

Then, $\frac{p}{q}=\frac{4}{11}$.

Therefore, $p+q=4+11$

$\Rightarrow p+q=15$.

Hence, $\frac{p}{q}=\frac{4}{11}$ and $p+q=15$.