How to Convert Decimals that Don’t Start Repeating Right Away Into Fractions

There are some decimals that repeat the same number or pattern of numbers forever…but they don’t start repeating until some number of digits after the decimal point. How can you convert this odd sort of repeating decimal into a fraction? The trick is to multiply the repeating decimal by two different powers of 10 so that the repeating decimal part goes away when you subtract the two numbers. The result of that subtraction (which is always just the non-repeating part of the repeating decimal) turns out to be the numerator of the resulting fraction. And how about the denominator? Well, that’s just the larger power of 10 minus the smaller power of 10.

Here’s what I mean: To convert 0.7222… into a fraction, start by multiplying 0.7222… by 10 to get 10 x 0.7222… = 7.222… and then by 100 to get 100 x 0.7222… = 72.222… Now subtract these two numbers to get 72.222… – 7.222… = 65. The repeating decimal goes away! And the number you get, 65, is going to be the numerator in our final answer. As you’ll recall, we multiplied the repeating decimal 0.7222… by 10 and 100 above, which means that the denominator of the fraction in the final answer must be 100 – 10 = 90. Which means that 0.7222… = 65/90 = 13/18.
For more, see How To Convert Repeating Decimals to Fractions, Part 3

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