Suppose you have some repeating decimal like
x = 0.123123123...
The repeating part has a length of 3, and multiplying both sides by 10^3 = 1000 gives
1000x = 123.123123123...
The usefulness of this is that we can subtract x from this to eliminate the decimal part:
1000x - x = 123.123... - 0.123...
999x = 123
then solve for x as a fraction:
x = 123/999
and in this case the fraction reduces further to
x = 41/333
