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What is

0.777...

when written as a fraction?

Answer

We can write this number as

This is a geometric series with and . The sum of the infinite series is

If we put into a calculator, we get something like

0.77777777778.

Since the calculator can only display finitely many digits, it rounds the last seven up to an 8.

Write

0.787878...

as a rational number.

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.

Make it rain.