Calculate
Give your answer as a fraction in its lowest terms.

Let’s begin by denoting the repeating decimals in a more manageable form.

Let (which means ), and let (which means ).

To convert into a fraction:

Multiplying both sides by 10, we have:

Subtracting the original from this equation:

Thus,

To convert into a fraction:

Multiplying both sides by 100 (because the repeating block is 2 digits long):

Next, we subtract the original :

Thus,

Now we need to compute :

To add these fractions, we need a common denominator. The least common multiple of 99 and 9 is 99. We will convert into a fraction with a denominator of 99:

Now we can add the two fractions:

Next, we can simplify :

Thus, the final answer is: