How to Divide Fractions by Fractions: 4 Easy Steps

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Plus example problems & solutions for dividing fractions
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Dividing a fraction by a fraction might seem confusing at first, but it is really very simple. All you need to do is flip the second fractions, multiply, and reduce! This article will guide you through the process , give you sample problems to solve, and show you that dividing fractions by fractions is really a breeze. We’ve also spoken to academic tutor David Jia and math teacher Grace Imson for their expert advice on dividing fractions by each other.

Solving for a Fraction Divided by a Fraction

Flip the numerator and denominator on the second fraction to get the reciprocal, says math teacher Grace Imson. Then, multiply the two numerators together to get the numerator of your final answer. Then, multiply the denominators together to get the denominator of your final answer. Simplify the fractions, if needed.

Section 1 of 4:

How to Divide a Fraction by a Fraction

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  1. You should have two fractions. The first fraction is the fraction that’s being divided—this is called your dividend fraction . The second fraction is the one that’s being divided by —this is called your divisor fraction . [1]
    • Example:
      • For our example problem, we’ll use 2/3 ÷ 3/7 .
      • 2/3 is our dividend fraction, while 3/7 is our divisor fraction.
  2. Your divisor fraction (or the second fraction) has a numerator at the top and a denominator at the bottom. Flip the fraction so your denominator is now at the top and the numerator at the bottom. This is called finding the reciprocal of your fraction . [2]
    • Example:
      • 3/7 is our divisor fraction. If we flip the fraction, the reciprocal is 7/3.
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  3. In order to divide fractions, we actually want to multiply them ! First, multiply the numerators of your two fractions together—the result of this calculation will become the numerator of your final answer. Then, multiply the denominators of your two fractions together—the result of this calculation will become the denominator of your final answer. [3]
    • Example:
      • 2 x 7 = 14 , so our numerator is 14.
      • 3 x 3 = 9 , so our denominator is 9.
      • This means that our final result is 14/9.
  4. If your fraction is already in its simplest form (e.g., 1/2, 2/3, 3/4, 5/7, etc.), then you’re done! If your fraction can still be simplified, however, then you must reduce it to its simplest form. To simplify a fraction , find the greatest common factor between the numerator and denominator (i.e., the largest number that both of them can be divided by), instructs Jia. Then, divide both the numerator and denominator by the greatest common factor. The resulting numbers are your new numerator and denominator in the simplest form of your fraction. [4]
    • In common terms, clarifies Jia, you just have to ask yourself “what is the biggest number that both the top and bottom can be divided by? What’s the biggest number that both numbers can be divided by?” [5]
    • Example:
      • In our example, the final result has a numerator (14) that’s larger than the denominator (9), meaning that we have to convert the fraction to a mixed number (a whole number and fraction combined, like 1⅔) before simplifying it.
      • We’ll first divide the numerator (14) by the denominator (9). 9 goes into 14 one time, with a remainder of 5, giving us this fraction → 1 5/9 (“one and five-ninths”).
      • That’s our answer! The fraction can’t be reduced further because the denominator is not evenly divisible by the numerator (i.e., 9 can’t be evenly divided by 5) and the numerator is a prime number (an integer that can only be divided by one and itself).
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Section 2 of 4:

Example Problems & Solutions

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  1. Solve 1/3 ÷ 2/5 = ____.
    • Solution:
      • 1/3 is our dividend and 2/5 is our divisor.
      • Flip 2/5 to find its reciprocal → 5/2.
      • Multiply together the numerators of each fraction → 1 x 5 = 5.
      • Multiply together the denominators of each fraction → 3 x 2 = 6.
      • Assemble your new fraction → 5/6.
      • This fraction can’t be simplified further, so you have your final answer!
  2. Solve 4/5 ÷ 2/6 = ____.
    • Solution:
      • 4/5 is our dividend and 2/6 is our divisor.
      • Flip 2/6 to find its reciprocal → 6/2.
      • Multiply together the numerators of each fraction → 4 x 6 = 24.
      • Multiply together the denominators of each fraction → 5 x 2 = 10.
      • Assemble your new fraction → 24/10.
      • To simplify, divide the numerator by the denominator → 24 ÷ 10 = 2 with a remainder of 4.
      • Write the answer out as a mixed fraction → 2 4/10 (two and four-tenths).
      • The greatest common factor in 4 and 10 is 2, so divide both numbers by 2 → 4 ÷ 2 = 2. 10 ÷ 2 = 5.
      • Write out these numbers in your final fraaction → 2 ⅖ (two and two-fifths).
  3. 3
    Example Problem #3 Solve 1/2 ÷ 1/6 = ____.
    • Solution:
      • 1/2 is our dividend and 1/6 is our divisor.
      • Flip 2/6 to find its reciprocal → 6/1.
      • Multiply together the numerators of each fraction → 1 x 6 = 6.
      • Multiply together the denominators of each fraction → 2 x 1 = 2.
      • Assemble your new fraction → 6/2.
      • To simplify, divide the numerator by the denominator → 6 ÷ 2 = 3.
      • This number can’t be simplified further, so you have your final answer!
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Section 3 of 4:

Understanding the Math Behind Dividing Fractions

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  1. The problem 2 ÷ 1/2 is asking you: ”How many halves are in 2?” The answer is 4, because each unit (1) is made up of two halves, and there are 2 units total: 2 halves/1 unit * 2 units = 4 halves. [6]
    • Try thinking about this same equation in terms of cups of water: How many half cups of water are in 2 cups of water? You could pour 2 half cups of water into each cup of water, which means you are basically adding them, and you have two cups: 2 halves/1 cup * 2 cups = 4 halves.
    • All of this means that when the fraction you are dividing by is between 0 and 1, the answer will always be larger than the original number! This is true whether you are dividing whole numbers or fractions by a fraction.
  2. That’s why dividing by a fraction can be accomplished by multiplying by its reciprocal (or its opposite ), explains Imson. [7] The reciprocal of a fraction (also called its “multiplicative inverse”) is just the fraction turned upside down, so that the numerator and denominator have switched places. Here are some examples of reciprocal fractions: [8]
    • The reciprocal of 3/4 is 4/3.
    • The reciprocal of 7/5 is 5/7.
    • The reciprocal of 1/2 is 2/1, or 2.
  3. Try remembering the following rhyme to help you remember: "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify, before it's time to say goodbye."
    • Another helpful saying that tells you what to do with each part of the equation is: “Leave Me (the first fraction) , Change Me (the division to a multiplication symbol) , Turn Me Over (the second fraction). ”
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Section 4 of 4:

More Math Lessons on Fractions

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  1. Fractions can be a fun but challenging part of any math class—but with the help of our numerous guides on fractions, you’re sure to ace every test! If you still have more to learn about how to work with and solve fractions, check out these articles:

Community Q&A

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  • Question
    What is 2 1/2 divided by 1/2?
    Donagan
    Top Answerer
    Change the mixed number to an improper fraction: 2½ becomes 5/2. Then invert ½ to become 2/1. Multiply 5/2 by 2/1, and reduce or simplify if possible. (5/2)(2/1) = 10/2 = 5.
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  • Question
    How do I divide 3/4 by 3/5?
    Donagan
    Top Answerer
    Invert 3/5 to 5/3, and multiply by ¾: (3/4) x (5/3) = 15/12 = 5/4 or 1¼.
    Thanks! We're glad this was helpful.
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    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
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  • Question
    How do I find the reciprocal of a whole number?
    Donagan
    Top Answerer
    The reciprocal is a fraction whose numerator is 1 and whose denominator is the whole number. For example, if the whole number is 5, think of it as 5/1, then invert that fraction to 1/5.
    Thanks! We're glad this was helpful.
    Thank you for your feedback.
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      References

      1. ↑ http://www.amathsdictionaryforkids.com/qr/d/division.html
      2. ↑ https://opentextbc.ca/mathfortrades1/chapter/dividing-fractions/
      3. ↑ https://resourcecenter.byupathway.edu/math/fractions/dividing-fractions
      4. ↑ David Jia. Academic Tutor. Expert Interview. 7 January 2021.
      5. ↑ David Jia. Academic Tutor. Expert Interview. 7 January 2021.
      6. ↑ https://thirdspacelearning.com/blog/how-to-divide-fractions/
      7. ↑ Grace Imson, MA. Math Instructor, City College of San Francisco. Expert Interview. 1 November 2019.
      8. ↑ https://opentextbc.ca/mathfortrades1/chapter/dividing-fractions/

      About This Article

      Article Summary X

      To divide fractions by fractions, start by replacing the division sign with a multiplication sign. Then, flip the second fraction over so the bottom number of the second fraction is now on the top. Multiply the top numbers of both fractions together to get the numerator (top number) of your new fraction. To get the denominator (bottom number) of your new fraction, multiply the bottom numbers of both fractions together. Simplify your fraction and you're finished! For actual examples of fractions being divided, read the article!

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