Convert Decimals with Repeating Patterns of Numbers Into Fractions

Learn Math Dude's super-easy conversion trick.

Jason Marshall, PhD
1-minute read

Convert Decimals with Repeating Patterns of Numbers Into Fractions

Any decimal that repeats some pattern of digits forever can be converted into a fraction that has the pattern of digits doing the repeating in its numerator and a denominator containing the same number of 9s as there are digits in the numerator. It sounds confusing, but it’s actually pretty easy. For example, in the repeating decimal 0.818181…, the pattern of digits “81” repeats over and over again after the decimal point.

According to the rule above, this decimal must be equal to the fraction that has 81 in its numerator. And since the number 81 has two digits, the denominator of this fraction must be the number made of two 9s…namely the number 99. So the decimal 0.818181… = 81/99. If we divide both the top and bottom by 9, we get the fully reduced answer: 0.818181… = 9/11.

For more, see How To Convert Repeating Decimals to Fractions, Part 2

About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

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