Question

How do you convert a repeating decimal to a fraction?

Answer 1

Count how many digits there are in the recurring part. Then write down as many 9s as the denominator and the recurring part as the numerator. No decimal points are needed. Example: 0.285714285714. There are 6 digits in the recurring part, so the denominator is 999999. Now we write 285714/999999=2/7. Example: 0.270270270. There are 3 digits in the recurring part, so we use 999 as the denominator, and the fraction is 270/999=10/37.
But what about decimals like 8.29230769230769? We move the decimal point to the right so that the recurring part starts immediately after the decimal point: 82.923076923076. In this case we moved the decimal place 1 place to the right. The fraction is 923076/999999=12/13. So the complete number is 82 12/13. Convert this to an improper fraction: 1078/13. Remember we moved the decimal point one place to the right so 1078/13 is 10 times too big, so we divide by 10=1078/130=539/65=8 19/65. This is the mixed number corresponding to the decimal. If we had to move the decimal point two places to the right, we would have divided by 100, so the number of zeroes after the 1 is the number of decimal places we shifted.