How to Multiply Mixed Fractions

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A quick and easy guide to multiplying mixed fractions with any type of number
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A mixed number is a whole number next to a fraction, such as 3 ½. Multiplying two mixed numbers can be tricky because you have to convert them to improper fractions first. If you want to know how to multiply mixed numbers by fractions, whole numbers, or other mixed numbers, you’ve come to the right place! You can learn how to multiply mixed fractions in just a few easy steps.

Multiplying Mixed Numbers Together

  1. Multiply each mixed number’s whole number by their fraction’s denominator.
  2. Add that product to the numerator and place it over the original denominator.
  3. Once you have two improper fractions instead of mixed numbers, multiply them.
  4. Multiply the numerators, then multiply the denominators to create a new fraction.
  5. Divide the numerator and denominator by their greatest common factor.
  6. Convert the fraction into a mixed number, if required.
Section 1 of 5:

How to Multiply Two Mixed Numbers

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  1. For each mixed number, multiply the whole number by the denominator of the fraction. Then, add that number to the numerator of the fraction. Place this new sum over the original denominator of the fraction to get an improper fraction equivalent to your original mixed number. [1]
    • For example , let’s say your first mixed number is 4½.
      • Multiply the whole number (4) by the fraction’s denominator (2) → 4 x 2 = 8 .
      • Add that number (8) to the fraction’s numerator (1) → 8 + 1 = 9 .
      • Replace the original numerator with this sum to get your improper fraction → 9 / 2 .
    • Repeat these steps with your second mixed number. In our example, the second mixed number will be 6⅖, which converts to the improper fraction 32 / 5 .
    • An improper fraction is a number whose numerator is greater than its denominator.
  2. Multiply the numerators by each other to get a new numerator. Then, multiply the denominators by each other to get a new denominator. Place the new numerator over the new dominator to get your final product. [2]
    • For example , our improper fractions are 9 / 2 and 32 / 5 .
    • Multiply the two numerators → 9 x 32 = 288 .
    • Multiply the two denominators → 2 x 5 = 10 .
    • Place the new numerator over the new denominator → 288 / 10 .
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  3. Find the Greatest Common Factor (GCF) between your numerator and denominator. The GCF is the biggest number by which both the numerator and denominator can be divided. Once you’ve found the GCF, divide both the numerator and denominator by that number. [3]
    • For example , our fraction is 288 / 10 .
    • The greatest common factor of both 288 and 10 is 2 .
    • Divide both numbers by their GCF → 288 / 2 = 144 . 10 / 2 = 5 .
    • Place your reduced numbers into their fraction → 144 / 5 .
    • The reduced form of 288 / 10 is 144 / 5 .
  4. Divide the numerator by the denominator by doing long division , using a calculator or dividing in your head. Write down the whole number part of the quotient—this is your new whole number. Take the remainder and write it over the original denominator. You now have the completed product of multiplying your mixed numbers. [4]
    • For example , take the improper fraction 144 / 5 .
    • Divide the numerator (144) by the denominator (5) to find the quotient and remainder .
    • 5 goes 28 times into 144. Therefore, the quotient is 28 . The remainder , or the number that’s left over, is 4 .
    • The original denominator was 5. Therefore, the final mixed fraction is 28⅘ . Finished!
    • Unless otherwise instructed, your final answer should be in mixed number form, just like the original question. If your teacher or workbook says that your final answer should be an improper fraction, you can skip this step.
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Section 2 of 5:

How to Multiply a Mixed Number by a Fraction

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  1. Multiply the whole number by the denominator of the fraction. Add that product to the numerator of the fraction. Use this new sum as a numerator over the original denominator of the fraction. You now have an improper fraction equal to your original mixed fraction. [5]
    • For example , let’s say your mixed number is 3½.
      • Multiply the whole number (3) by the fraction’s denominator (2) → 3 x 2 = 6 .
      • Add that number (6) to the fraction’s numerator (1) → 6 + 1 = 7 .
      • Replace the original numerator with this sum to get your improper fraction → 7 / 2 .
  2. Now that your mixed fraction has been converted into an improper fraction, you can multiply it by any other fraction. Multiply the numerators together, then multiply the denominators. Place the product of the numerators over the product of the denominators to get a new fraction. [6]
    • For example , let’s say you are multiplying 7 / 2 by ¾.
    • Multiply the two numerators → 7 x 3 = 21 .
    • Multiply the two denominators → 2 x 4 = 8 .
    • Place the new numerator over the new denominator → 21 / 8 .
  3. Find the greatest common factor (GCF) between your numerator and denominator. If the GCF (the largest number both can be divided by) is greater than 1, then you can simplify your fraction. Divide the numerator and the denominator by the GCF to get a simplified version of your fraction. [7]
    • The fraction 21 / 8 is already simplified because the GCF of 21 and 8 is 1.
    • If the fraction was 20 / 8 , however, the GCF of 10 and 8 is 4. If you divide both 20 and 8 by their GCF (4), you get the simplified version of this fraction → 5 / 2 .
  4. Divide the numerator of the fraction by the denominator. However many times your numerator goes into your denominator is the quotient. The whole number of the quotient becomes the whole number of your new mixed number. The number left over after the quotient is called the remainder, which becomes your new numerator. Write your numerator over the original denominator to find your final answer. [8]
    • For example , take the improper fraction 21 / 8 .
    • Divide the numerator (21) by the denominator (8) to find the quotient and remainder .
    • 8 goes 2 times into 21. Therefore, the quotient is 2 . The remainder , or the number that’s left over, is 5 .
    • The original denominator was 8. Therefore, the final mixed fraction is 2⅝ . Aced it!
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Section 3 of 5:

How to Multiply a Mixed Fraction by a Whole Number

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  1. Multiply the whole number by the fraction’s denominator (or bottom number). Take that calculated number and add it to the fraction’s original numerator. Place this sum over the denominator of the original fraction. This new fraction is the converted version of your original mixed number. [9]
    • For example , let’s say your mixed number is 3⅖.
      • Multiply the whole number (3) by the fraction’s denominator (5) → 3 x 5 = 15 .
      • Add that number (15) to the fraction’s numerator (2) → 15 + 2 = 17 .
      • Replace the original numerator with this sum to get your improper fraction → 17 / 5 .
  2. Whole numbers are secretly hidden fractions with a denominator of 1. The whole number 2, for example, can be rewritten as the fraction 2 / 1 , and the same is true of any whole number. Place the whole number that you’re multiplying your mixed number by as a numerator over the denominator 1. [10]
    • For example , let’s say you’re multiplying the mixed number 3⅖ by the whole number 4.
    • Place 4 as a numerator over the denominator 1 → 4 / 1 .
  3. You now have two improper fractions that must be multiplied together. Multiply the numerators to get a new numerator. Then, multiply the denominators to get a new denominator. Find the product of the two fractions by placing the new numerator on top of the new denominator. [11]
    • For example , take the fractions 17 / 5 and 4 / 1 .
    • Multiply the two numerators → 17 x 4 = 68 .
    • Multiply the two denominators → 5 x 1 = 5 .
    • Place the new numerator over the new denominator → 68 / 5 .
  4. If your fraction can be further reduced, find the greatest common factor (GCF) between your numerator and denominator or the biggest number by which both are divisible. Divide both the numerator and denominator by the GCF to get a simplified form of your fraction. [12]
    • The improper fraction 68 / 5 is already in its most simplified form because the biggest number that both 68 and 5 can be divided by is 1.
  5. Divide the numerator of your fraction by the denominator. Once you get the quotient, note its whole number and remainder. The whole number is the whole number of your new mixed fraction. The remainder is your new numerator, which you write over the original denominator to get your final answer. [13]
    • For example , take the improper fraction 68 / 5 .
    • Divide the numerator (68) by the denominator (5) to find the quotient and remainder .
    • 5 goes 13 times into 68. Therefore, the quotient is 13 . The remainder , or the number that’s left over, is 3 .
    • The original denominator was 5. Therefore, the final mixed fraction is 13⅗ . Voila!
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Section 4 of 5:

What are mixed numbers?

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  1. Mixed numbers (or mixed fractions) are made up of three parts: a whole number, a numerator, and a denominator. The numerator and denominator form the proper fraction, which combines with the whole number to create a mixed number. [14]
    • For example, the mixed number 2⅓ is a combination of an integer or whole number (2) and a proper fraction (⅓). The proper fraction (⅓) is made up of a numerator (1) and a denominator (3).
    • Mixed numbers generally represent a number between two whole numbers. The mixed number 2⅓, for example, is greater than 2 but less than 3. Therefore, it’s a mixed number between two whole numbers.
    • The fractional part of the mixed number must be a proper fraction. In other words, it has to be less than one whole, and the numerator (top number) must be greater than the denominator (bottom number).
    • Mixed fractions are always greater than 1. Fractions like ½ or ⅛ are not mixed numbers because they are less than 1 and aren’t represented with a whole number.
Section 5 of 5:

Example Problems & Solutions

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  1. Multiply 2⅕ x 4½. Simplify the answer and write as a mixed number. [15]
    • Solution:
      • Convert 2⅕ to an improper fraction → 11 / 5 .
      • Convert 4½ to an improper fraction → 9 / 2 .
      • Rewrite the multiplication problem with the improper fractions: 11 / 5 x 9 / 2 .
      • Multiply the numerators and denominators → 11 x 9 / 5 x 2 = 99 / 10 .
      • Rewrite the fraction as a mixed number → 99 / 10 = 9 9 / 10 .
  2. Multiply ½ x 3⅓. Simplify the answer and write as a mixed number. [16]
    • Solution:
      • Change 3⅓ to an improper fraction → 10 / 3 .
      • Rewrite the multiplication problem with the improper fraction in place of the mixed number: ½ x 10 / 3 .
      • Multiply the numerators and multiply the denominators → 1 x 10 / 2 x 3 = 10 / 6 .
      • Simplify the fraction by using its common factor (2) → 10 / 6 = 5 / 3 .
      • Rewrite the fraction as a mixed number → 5 / 3 = 1⅔ .
  3. Tim ate 3⅔ pizzas in a week. Susie ate 2¼ times that amount in a week. How much pizza did Susie eat? [17]
    • Solution:
      • Change each mixed number to an improper fraction: 3⅔ = 11 / 3 . 2¼ = 9 / 4 .
      • Multiply the fractions: 11 x 9 = 99 . 3 x 4 = 12 . 11 / 3 x 9 / 4 = 99 / 12 .
      • Simplify the fraction, if possible: 99 / 12 → 33 / 4 .
      • Convert to a mixed number, if possible: 33 / 4 → Susie ate 8¼ pizzas.
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Mixed Numbers Multiplication Worksheet

References

  1. ↑ https://www.mathsisfun.com/improper-fractions.html
  2. ↑ https://www.cuemath.com/numbers/multiplying-fractions/
  3. ↑ https://www.mathsisfun.com/greatest-common-factor.html
  4. ↑ https://www.cuemath.com/numbers/improper-fraction-to-mixed-number/
  5. ↑ https://www.mathsisfun.com/improper-fractions.html
  6. ↑ https://www.cuemath.com/numbers/multiplying-fractions/
  7. ↑ https://www.mathsisfun.com/greatest-common-factor.html
  8. ↑ https://www.cuemath.com/numbers/improper-fraction-to-mixed-number/
  9. ↑ https://www.mathsisfun.com/improper-fractions.html
  1. ↑ https://www.geeksforgeeks.org/how-to-convert-a-whole-number-to-an-improper-fraction/
  2. ↑ https://www.cuemath.com/numbers/multiplying-fractions/
  3. ↑ https://www.mathsisfun.com/greatest-common-factor.html
  4. ↑ https://www.cuemath.com/numbers/improper-fraction-to-mixed-number/
  5. ↑ https://thirdspacelearning.com/us/blog/what-is-mixed-number/
  6. ↑ https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/02%3A_Fractions_and_Mixed_Numbers/2.02%3A_Multiplying_and_Dividing_Fractions_and_Mixed_Numbers/2.2.01%3A_Multiplying_Fractions_and_Mixed_Numbers
  7. ↑ https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/02%3A_Fractions_and_Mixed_Numbers/2.02%3A_Multiplying_and_Dividing_Fractions_and_Mixed_Numbers/2.2.01%3A_Multiplying_Fractions_and_Mixed_Numbers
  8. ↑ https://oryxlearning.com/learn/multiplying-mixed-numbers-word-problems/

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  • Question
    When simplifying, is it the same way as doing it with any other operation?
    Community Answer
    In terms of fractions, yes. You find the greatest common factor, then you factor the fraction/mixed number.
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    How do I divide mixed numbers?
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    Turn each mixed number into an improper fraction, then divide as you would with any fractions.
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    What if after Imultiply the improper fraction, the numbers I get don't have any factors in common?
    LyKaxandra Caimoy
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    If you are referring to reducing it to lowest term, then that is the simplest form or, otherwise, convert it to mixed numbers.
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      To multiply mixed numbers, start by converting each mixed number to an improper fraction. Then, multiply the improper fractions together. Reduce your answer to the lowest terms using the greatest common factor. Finally, convert your answer back to a mixed number. To learn how to find the greatest common factor to reduce your fraction, keep reading the article!

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