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

Answer 1

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.

Answer 2

To convert 0.77777777... into a fraction, we first denote it as x. Next, we multiply x by 10 to shift the decimal point to the right, yielding 10x = 7.77777777... Subtracting x from 10x gives 9x = 7, which simplifies to x = 7/9. Therefore, 0.77777777... is equivalent to the fraction 7/9.