# Question #a1e8f

Jan 6, 2017

$0.6 \overline{12} = \frac{101}{165}$

#### Explanation:

Let $x = 0.6 \overline{12}$

Note that the repeating portion of the decimal is $2$ digits long, so we multiply $x$ by ${10}^{2} = 100$.

$100 x = 61.2 \overline{12}$

$\implies 100 x - x = 61.2 \overline{12} - 0.6 \overline{12}$

$\implies 99 x = 61.2 - 0.6$

$\implies 99 x = 60.6$

$\implies 99 x = \frac{606}{10}$

$\implies x = \frac{606}{10} \cdot \frac{1}{99}$

$\therefore x = \frac{101}{165}$

For a detailed approach to general questions of this type, see this question.

Jan 6, 2017

$\frac{101}{165}$

#### Explanation:

Note that $0.6 \overline{12}$ is the same as $0.612121212 \ldots$

Let $x = 0.6 \overline{12}$

Then $10 x = 6.121212 \ldots$

and $1000 x = 612.121212 \ldots$

So $1000 x - 10 x = 612.121212 \ldots$
$\text{ "ul(color(white)(61)6.121212...) larr" subtract}$
$\text{ } 1000 x - 10 x = 606.000000 \ldots . .$

$\implies x \left(1000 - 10\right) = 990 x = 606$

$\implies x = \frac{606}{990} = \frac{101}{165}$