Write

0.787878...

as a rational number.

Answer

This is very similar to the previous problem. We use the repeating pattern to break up the decimal into a sum of fractions:

Since the repeating pattern is 2 digits long, we have to multiply each fraction by to get to the next fraction. We have a geometric series with and .

The sum of this series is

If we put the fraction into a calculator, we get something very close to 0.787878.... This is a good sanity check.