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How to Convert Repeating Decimals to Fractions Download Article Quickly change repeating decimals to simple fractions with easy math Reviewed by Grace Imson, MA
Last Updated: December 2, 2024 Fact Checked
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Basic Repeating Decimals
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Mixed Repeating & Non-Repeating Decimals
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This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 184,169 times.
A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or digits that infinitely repeat at regular intervals.[1]XResearch source Repeating decimals can be tricky to work with, but they can also be converted into a fraction. Sometimes, repeating decimals are indicated by a line over the digits that repeat. The number 3.7777 with 7 repeating, for instance, can also be written as 3.7. To convert a number like this to a fraction
you write it as an equation, multiply, subtract to remove the repeating decimal, and solve the equation.
Steps
Part 1 Part 1 of 2:
Converting Basic Repeating Decimals
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1 Locate the repeating decimal. For instance, the number 0.4444 has a repeating decimal of 4. It is a basic repeating decimal in the sense that there's no non-repeating portion to the decimal number. Count how many repeating digits there are in the pattern.
Once your equation is written, you will multiply it by 10^y, where y equals the number of repeating digits in the pattern.[2]XResearch source
In the example of 0.4444, there is one digit that repeats, so you will multiply the equation by 10^1.
For a repeating decimal of 0.4545, there are two digits that repeat, and you would, therefore, multiply your equation by 10^2.
For three repeating digits, multiply by 10^3, etc.
2 Rewrite the decimal as an equation. Write it out so that x equals the original number. [3]XResearch source In this instance, the equation is x = 0.4444. Since there’s only one digit in the repeating decimal, multiply the equation by 10^1 (which equals 10).[4]XResearch source
In the example where x = 0.4444, then 10x = 4.4444.
With the example x = 0.4545, there are two repeating digits, so you multiply both sides of the equation by 10^2 (which equals 100), giving you 100x = 45.4545.
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3 Remove the repeating decimal. You accomplish this by subtracting x from 10x. Remember that whatever you do to one side of the equation must be done to the other, so:[5]XResearch source
10x – 1x = 4.4444 – 0.4444
On the left side, you have10x - 1x = 9x. On the right side, you have 4.4444 – 0.4444 = 4
Therefore, 9x = 4
4 Solve for x. Once you know what 9x equals, you can determine what x equals by dividing both sides of the equation by 9:
On the left side of the equation you have 9x ÷ 9 = x. On the right side of the equation you have 4/9
Therefore, x = 4/9, and the repeating decimal 0.4444 can be written as the fraction 4/9.
5 Reduce the fraction. Put the fraction in its simplest form (if applicable) by dividing both the numerator and denominator by the greatest common factor.[6]XResearch source
In the example of 4/9, that is the simplest form.
EXPERT TIP
Joseph Meyer
Math Teacher Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. Joseph Meyer Math Teacher
To simplify fractions, you can divide both the numerator and denominator by a common factor. This creates a new, easier-to-use fraction with smaller components, but it represents the same value. For instance, if you divide both the numerator and denominator of 6/12 by 2, you get 3/6, which is equal to 1/2.
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Converting Numbers With Repeating and Non-Repeating Decimals
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1 Determine the repeating digits. It’s not uncommon for a number to have non-repeating digits before the repeating decimal, but these can still be converted into fractions.[7]XResearch source
For example, take the number 6.215151. Here, 6.2 is non-repeating, and the repeating digits are 15.
Again take note of how many repeating digits there are in the pattern, because you will multiply by 10^y based on that number.
In this example, there are two repeating digits, so you will multiply your equation by 10^2.
2 Write the problem as an equation and subtract the repeating decimals. Again, if x = 6.215151, then 100x = 621.5151. To remove the repeating decimals, subtract from both sides of the equation:[8]XResearch source
100x – x (= 99x) = 621.5151 - 6.215151 (= 615.3)
Therefore, 99x = 615.3
3 Solve for x. Since 99x = 615.3, divide both sides of the equation by 99. This gives you x = 615.3/99.
4 Remove the decimal in the numerator. Do this by multiplying the numerator and denominator by 10^z, where z equals the number of decimal places you must move to eliminate the decimal.[9]XResearch source In 615.3, you have to move the decimal by one place, meaning you multiply the numerator and denominator by 10^1:
615.3 x 10 / 99 x 10 = 6153/990
Reduce the fraction by dividing the numerator and denominator by the highest common factor, which in this case is 3, giving you x = 2,051/ 330[10]XResearch source
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Question How do I make an improper fraction into a mixed number? Donagan Top Answerer To convert an improper fraction to a mixed number, divide the denominator into the numerator. The whole number of the quotient is the whole number of the mixed number. If the quotient also has a remainder, the remainder is the numerator of the fraction in the mixed number. The denominator of the fraction is the same as the denominator of the improper fraction. For example, to convert 13/5 to a mixed number, divide 5 into 13. The quotient is 2-3/5. 2 is the whole number of the mixed number. 3 is the numerator of the fraction of the mixed number. 5 is the denominator of the fraction of the mixed number. Thus, the mixed number is 2-3/5 (two and three-fifths). Thanks! We're glad this was helpful.Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 5 Helpful 18
Question How do I solve a cubic equation? Donagan Top Answerer See Solve a Cubic Equation. Thanks! We're glad this was helpful.Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 8 Helpful 14
Question How do I show that the reciprocal of 0.131313 is 7.5? Donagan Top Answerer Actually it's not. Use a calculator to divide 1 by 0.131313. You'll get 7.615. Thanks! We're glad this was helpful.Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 19 Helpful 25
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Reviewed by: Grace Imson, MA Math Instructor, City College of San Francisco This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 184,169 times. 41 votes - 68% Co-authors: 11Updated: December 2, 2024 Views: 184,169 Categories: Fractions | Conversion AidsArticle SummaryX
To convert repeating decimals to fractions, start by writing an equation where x equals your original number. For example, x = 0.4444. Then, multiply both sides of the equation by 10^1, since there’s just 1 repeating digit in your original number, to get 10x = 4.4444. Next, remove the repeating decimal by subtracting x from 10x on both sides to get 9x = 4. Finally, solve for x to get 4/9. To learn how to convert numbers with repeating and non-repeating decimals, scroll down!Did this summary help you?YesNo
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Download Article Quickly change repeating decimals to simple fractions with easy math Reviewed by Grace Imson, MA 
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Reviewed by: Grace Imson, MA Math Instructor, City College of San Francisco This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 184,169 times. 41 votes - 68% Co-authors: 11 Updated: December 2, 2024 Views: 184,169 Categories: Fractions | Conversion Aids Article SummaryX
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1 Locate the repeating decimal. For instance, the number 0.4444 has a repeating decimal of 4. It is a basic repeating decimal in the sense that there's no non-repeating portion to the decimal number. Count how many repeating digits there are in the pattern. 
2 Rewrite the decimal as an equation. Write it out so that x equals the original number. [3] In this instance, the equation is x = 0.4444. Since there’s only one digit in the repeating decimal, multiply the equation by 10^1 (which equals 10).[4] 
3 Remove the repeating decimal. You accomplish this by subtracting x from 10x. Remember that whatever you do to one side of the equation must be done to the other, so:[5] 
4 Solve for x. Once you know what 9x equals, you can determine what x equals by dividing both sides of the equation by 9: 
5 Reduce the fraction. Put the fraction in its simplest form (if applicable) by dividing both the numerator and denominator by the greatest common factor.[6] 
1 Determine the repeating digits. It’s not uncommon for a number to have non-repeating digits before the repeating decimal, but these can still be converted into fractions.[7] 
2 Write the problem as an equation and subtract the repeating decimals. Again, if x = 6.215151, then 100x = 621.5151. To remove the repeating decimals, subtract from both sides of the equation:[8] 
3 Solve for x. Since 99x = 615.3, divide both sides of the equation by 99. This gives you x = 615.3/99. 
4 Remove the decimal in the numerator. Do this by multiplying the numerator and denominator by 10^z, where z equals the number of decimal places you must move to eliminate the decimal.[9] In 615.3, you have to move the decimal by one place, meaning you multiply the numerator and denominator by 10^1: 
Donagan Top Answerer To convert an improper fraction to a mixed number, divide the denominator into the numerator. The whole number of the quotient is the whole number of the mixed number. If the quotient also has a remainder, the remainder is the numerator of the fraction in the mixed number. The denominator of the fraction is the same as the denominator of the improper fraction. For example, to convert 13/5 to a mixed number, divide 5 into 13. The quotient is 2-3/5. 2 is the whole number of the mixed number. 3 is the numerator of the fraction of the mixed number. 5 is the denominator of the fraction of the mixed number. Thus, the mixed number is 2-3/5 (two and three-fifths). Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 5 Helpful 18