Question

Represent the following mixed infinite decimal periodic fractions as common fractions
$2.1\overline{32}$

Answer 1

We can write it as $\mathrm{2.1323232323232....}$

Let $x=\mathrm{2.132323232....}$ $\left(1\right)$

Multiply $\left(1\right)$ by $1000$ on both sides, we get

$10\times x$ $=\mathrm{2.1323232........}\times 10$

$10x=\mathrm{21.3232323232323...}$ $\left(2\right)$

Now again multiply $100 both side in equation1

$10x\times 100$=$\mathrm{21.3232323232...}\times 100$

$1000x=\mathrm{2132.32323232.....}$ $\left(3\right)$

Now subtract $\left(2\right)$ from $\left(3\right)$ we have

$1000x-10x=2132.32323232-21.3232323232$

$990x=2111$

$\Rightarrow x=\frac{2111}{990}$