3.3.2 In the form: .ababa …

In a similar vein, fractions in the form $.ababab \dots$ can be treated as:

$$.ababab \dots = \frac{ab}{100} + \frac{ab}{10000} + \frac{ab}{1000000} + \dots = \frac{\frac{ab}{100}}{1 - \frac{1}{100}} = \frac{ab}{100} \times \frac{100}{99} = \frac{ab}{99}$$

Where $ab$ represents the digits (not $a \times b$). Here is an example:

$$.242424 \dots = \frac{24}{99} = \frac{8}{33}$$

You can extend the concept for any sort of continuously repeating fractions. For example, $.abcabcabc \dots = \frac{abc}{999}$, and so on.

Here are some practice problems to help you out:

Problem Set 3.3.2