Question

Express in simplest $\frac{p}{q}$ form and find the value $p+q$.

$0.3\overline{4}$

Answer 1

Let $x=0.3\overline{4}.................\left(1\right)10x=3.\overline{4}.................\left(2\right)100x=34.\overline{4}.................\left(3\right)$

subtracting (2) from (3)

$\Rightarrow 100x-10x=34.\overline{4}-3.\overline{4}\Rightarrow 90x=31\Rightarrow x=\frac{31}{90}$

Comparing $\frac{p}{q}=\frac{31}{90}p+q=31+90=121$