How can you convert 0.77777777... into a fraction without using a calculator?

First find how many digits repeat. So in this case one digit repeats. Then take that digit (or digits as a numerator). Now take as denominator 9, or 99, or 999, etc. (the same number of nines as repeating digits). So in this case it's just 1 nine.

So we have 7/9 is the fraction equivalent of 0.77777....

Here's another : 0.18181818.....

There are two repeating digits (18), so use 2 nines (99). We have 18/99 then simplify (2/11).

Here's one more: 0.142857142857142857..... There are six digits (142857), so we have (142857/999999) [six 9's], which simplifies to 1/7.