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Convert Decimals with Repeating Patterns of Numbers Into Fractions Learn Math Dude's super-easy conversion trick.

By
Jason Marshall, PhD, 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.

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