Pick a positive integer.
Divide the integer by a denominator that contains as many 9s as there are digits in the numerator. For example, if you pick 330, the fraction would be 330/999. The answer will be a repeating decimal (in this case, 0.330330330...). This pattern holds true for any integer.
Reduce the fraction. Divide the numerator and denominator by any factor that goes into both and leaves no remainder (common denominator). For example, 330/999 reduces to 110/333 because 3 goes into both 330 and 999 evenly.
Divide the reduced fraction. In the example, 110/333 gives a quotient of 0.330330330. The number is the same as the decimal in Step 2 because the reduced fraction is equal to the original fraction.
Apply the principle in reverse. Begin with a fraction with only 9s in the denominator, for example, 124/999. Determine its decimal equivalent: 0.124124124... .
Begin with a fraction with a factor of 9, 99 or 999 in the denominator. For example, 123/333. Predict its decimal equivalent, 0.369369369... . "Un-reduce" the fraction to 369/999 to get the answer.