Question

Express $0.4\overline{7}$ in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.

Answer 1

$0.4\overline{7}$= 0.4777...

$=\frac{4}{10}+\frac{0.777}{10}$

Let x = 0.777…

$\Rightarrow$ 10x = 7.777… [Since only one digit is repeating, multiply x with 10.]
$\Rightarrow$ 10x = 7+ 0.777...
$\Rightarrow$ 10x = 7 + x
$\Rightarrow$ $x=\frac{7}{9}$
Therefore,
$0.4\overline{7}=\frac{4}{10}+\frac{x}{10}=\frac{4}{10}+\frac{7}{90}=\frac{36+7}{90}=\frac{43}{90}$