Let's first assign X the value of the repeating decimal. So
X = 0.55555555......
Let's multiply both sides by 10
10X = 5.55555555.......
Now let's subtract the 1st equality from the 2nd.
10X = 5.55555555.....
- (X = 0.55555555.....)
=
9X = 5
And finally, divide both sides by 9.
X = 5/9
You can use the above technique for ANY repeating decimal by multiplying by 10^n power where n is the number of digits in the repeating cycle. For instance,
1/7 = 0.142857142857142857142857142857142857....
So the repeating portion is 142857 which is 6 digits long. So you'd multiply by 10^6 or 1000000.
