3.3.3 In the form: .abbbb …

Fractions in the form $.abbbb \dots$ are treated in a similar manner (sum of an infinite series) with the inclusion of one other term (the $.a$ term). Let’s see how it would look:

$$.abbb \dots = \frac{a}{10} + \frac{b}{100} + \frac{b}{1000} + \dots = \frac{a}{10} + \frac{\frac{b}{100}}{1 - \frac{1}{10}} = \frac{a}{10} + \frac{b}{90}$$

However we can continue and rewrite the fraction as:

$$\frac{a}{10} + \frac{b}{90} = \frac{9 \cdot a + b}{90} = \frac{(10 \cdot a + b) - a}{90}$$

Lets take a step back to see what this means. The numerator is composed of the sum $(10 \cdot a + b)$ which represents the two-digit number $ab$. Then you subtract from that the non-repeating digit and place that result over 90. Here is an example to show the process:

$$.27777 \dots = \frac{27 - 2}{90} = \frac{25}{90} = \frac{5}{18}$$

Here are some more problems to give you more practice:

Problem Set 3.3.3