How to Convert a Recurring Decimal Into a Fraction
A repeating decimal is a decimal whose digit or digits repeat over and over without end. Here are some examples of repeating decimals: .333333.…., .27272727.….. The ellipsis indicates that the numbers continue, and a bar can be drawn above the repeating number or numbers to indicate that those digits repeat continually. At first glance, it may seem difficult or even impossible to figure out how to convert a decimal like .272727... into a fraction; however, repeating decimals are rational numbers. That means, by definition, they can be represented as a fraction a/b, where a and b are integers and b is not equal to 0. The key is to use the formula for the sum of an infinite geometric sequence. Then, it’s simple to convert a recurring decimal into a fraction.
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Instructions
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Write your decimal as the sum of its fractional parts. For instance, .2727272... Becomes .27 + .0027 + .000027 + .00000027 + …
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Use the formula for the sum (S) of an infinite geometric sequence:S = a1 /(1-r). The first term of your sequence is a1 and r is the relationship between the terms of the sequence. In this example, r = .01 because the first term (.27) times .01 yields the second term (.0027); the second term times .01 yields the third term (.000027) and so on. The absolute value of r must be less than 1 ( lrl < 1).
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Plug your numbers into the formula. For our example, S = .27/ (1-.01) = .27/.99 = 27/99 = 3/11.Three-elevenths (3/11) is your answer.
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Use the same method even if your number begins as a non-repeating number. For instance, 5.13333333333. Writing this as the sum of its fractional parts yields51/10 + 3/100 + 3/1000 + 3/10000 + ……You should recognize that the repeating part is a geometric series and apply the formulaS = a1 /(1-r). S = (3/100)/(1-.1) = .03/.9 = 1/30. Add this to the non-repeating part of the number (51/10) and you get 51/10 + 1/30 = 154/30 = 77/15
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Tips & Warnings
Try this method out on common repeating decimals whose fractions you already know or can easily deduce, such as .66666.…. = 2/3, and 111111.… = 1/9 for practice.
Resources
- Photo Credit Kriss Szkurlatowski
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Comments
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can someone help me how to solve this? 1.7777, 2.565656, 3.181818.
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oh no no! thats not it i meant how do u write 1.19 while the nine repats as a fraction? oops! i said it wrong! (:
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Put a bar above the 9 - 9 Like that
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can u please tell me how to write 1.19 while the 9 repeats because I'm having some diffuclty with it thank u (:
