Question

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

$5.\overline{2}$

Answer 1

let $x=5.\overline{2}=\mathrm{5.22222222222222222222222222.................}\left(1\right)10x=52.\overline{2}=\mathrm{52.22222222222222222222222222.................}\left(2\right)$

subtracting (1) from (2)

$\Rightarrow 10x-x=52.\overline{2}-5.\overline{2}\Rightarrow 9x=47\Rightarrow x=\frac{47}{9}$

Comparing $\frac{p}{q}=\frac{47}{9}p+q=47+9=56$