Question

Express $0.6+0.4\overline{7}+0.\overline{7}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

Answer 1

$0.6+0.4\overline{7}+0.\overline{7}$ in $\frac{p}{q}$ form.

$0.6=\frac{6}{10}$

Let $x=0.4\overline{7}-----\left(1\right)$

Multiplying by $100$

$100x=47.\overline{7}-----\left(2\right)$

substract $\left(2\right)-\left(1\right)$

$100x-x=47.\overline{7}-0.4\overline{7}$

$99x=47.3$

$x=\frac{47.3}{99}$

Let $y=0.\overline{7}-----\left(3\right)$

Multiplying by $10$

$10y=7.\overline{7}-----\left(4\right)$

$\left(4\right)-\left(3\right)$

$10y-y=7.\overline{7}-0.\overline{7}$

$9y=7$

$y=\frac{7}{9}$

Now, $0.6+0.4\overline{7}+0.\overline{7}=\frac{6}{10}+\frac{47.3}{99}+\frac{7}{9}$

The LCM is $990$

$\frac{6}{10}+\frac{47.3}{99}+\frac{7}{9}=\frac{6}{10}\times \frac{99}{99}+\frac{47.3}{99}\times \frac{10}{10}+\frac{7}{9}\times \frac{110}{110}$

$\Rightarrow \frac{594}{990}+\frac{473}{990}+\frac{770}{990}$

$\Rightarrow \frac{594+473+770}{990}=\frac{1837}{990}$