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Answer 1 · 2017-01-06T22:30:31

$0.6 ¯ ¯¯ ¯ 12 = \frac{101}{165}$ $0.6bar(12) = 101/165$

Let $x = 0.6 ¯ ¯¯ ¯ 12$ $x = 0.6bar(12)$

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

$100 x = 61.2 ¯ ¯¯ ¯ 12$ $100x = 61.2bar(12)$

$\Rightarrow 100 x - x = 61.2 ¯ ¯¯ ¯ 12 - 0.6 ¯ ¯¯ ¯ 12$ $=> 100x-x = 61.2bar(12)-0.6bar(12)$

$\Rightarrow 99 x = 61.2 - 0.6$ $=> 99x = 61.2-0.6$

$\Rightarrow 99 x = 60.6$ $=> 99x = 60.6$

$\Rightarrow 99 x = \frac{606}{10}$ $=> 99x = 606/10$

$\Rightarrow x = \frac{606}{10} \cdot \frac{1}{99}$ $=> x = 606/10 * 1/99$

$:. x = 101/165$

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

Answer 2 · 2017-01-06T22:35:02

$101/165$

Note that $0.6bar(12)$ is the same as $0.612121212...$

Let $x=0.6bar(12)$

Then $10x=6.121212...$

and $1000x=612.121212...$

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

$=>x(1000-10)=990x=606$

$=>x=606/990 = 101/165$