Question

What are the easy ways to complete problems like these?

Answer 1

You can think of this whole process as "surgically removing" the repeating part from the decimal to get it into fraction form. I'll break down the steps:

$x=0.\overline{12}$

We set some variable x equal to our repeating decimal.

$100x=12.\overline{12}$

We want to move the decimal point just enough to the right so that we can make the "chop" we're going to perform in the next step. Since the decimal starts repeating every two digits, we want to scoot the decimal point two places over, which we can do by multiplying it by 100. We do the same to the x to keep both sides equal.

$100x-x=12.\overline{12}-0.\overline{12}$

This is the "chop." We're subtracting the original value of x from both sides to get rid of the repeating part of $12.\overline{12}$. On the left side, this leaves us with

$99x=12$

We can then divide either side of the equation by 99 to find that

$x=\frac{12}{99}=\frac{4}{33}$