How to Divide Mixed Fractions

Co-authored by David Jia

A mixed number, or mixed fraction, is a number that combines a whole number and a fraction. It is possible to divide mixed numbers; however, doing so requires converting them to improper fractions first. Once the mixed numbers are converted, you can divide as you would divide any other fractions.

Part 1

Part 1 of 3:

Converting Mixed Numbers to Improper Fractions

1
Multiply the whole number by the denominator of its combined fraction.^{[1]
X
Research source} Do this for both mixed numbers. Set these products aside. They are only part of your new numerator.
- For example, if you want to calculate $6{\frac {1}{2}}\div 2{\frac {1}{4}}$ , you would multiply $6\times 2=12$ and $2\times 4=8$ .
2
Add the numerator to the product.^{[2]
X
Research source}Do this for both mixed numbers. This sum will be the numerator of your improper fraction.
- For example, $12+1=13$ and $8+1=9$ .
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3
Place the sum over the original denominator.^{[3]
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Research source} Complete this step for both fractions, making sure you use the correct denominators. These are your improper fractions that you will use to complete the division.
- For example, $6{\frac {1}{2}}$ becomes ${\frac {13}{2}}$ and $2{\frac {1}{4}}$ becomes ${\frac {9}{4}}$ .
4
Convert whole numbers to fractions. If you are working with any whole numbers, you need to convert them to fractions. To do this, turn the number into the numerator of a fraction. The denominator will be 1.^{[4]
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Research source}
- For example, $3={\frac {3}{1}}$ .

Part 2

Part 2 of 3:

Dividing Improper Fractions

1
Write the new division problem. Use the improper fractions you found by completing the calculations in Part 1.
- For example, ${\frac {13}{2}}\div {\frac {9}{4}}$ .
2
Take the reciprocal of the second fraction.^{[5]
X
Research source} To find a reciprocal of a fraction, you need to “flip” it, so that the numerator becomes the denominator, and the denominator becomes the numerator.^{[6]
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Research source} Then, change the problem to a multiplication problem.^{[7]
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Research source}^{[8]
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Expert Source

David Jia
Academic Tutor
Expert Interview. 23 February 2021.}
- For example, if you take the reciprocal of ${\frac {9}{4}}$ , it becomes ${\frac {4}{9}}$ . So ${\frac {13}{2}}\div {\frac {9}{4}}$ becomes ${\frac {13}{2}}\times {\frac {4}{9}}$
3
Multiply the numerators. To do this, multiply them as if they were whole numbers. This product will be the numerator of your answer.^{[9]
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Research source}^{[10]
X
Expert Source

David Jia
Academic Tutor
Expert Interview. 23 February 2021.}
- For example, if calculating ${\frac {13}{2}}\times {\frac {4}{9}}$ , you would multiply the numerators: $13\times 4=52$ .
4
Multiply the denominators. To do this, multiply them as if they were whole numbers. This product will be the denominator of your answer.^{[11]
X
Research source}^{[12]
X
Expert Source

David Jia
Academic Tutor
Expert Interview. 23 February 2021.}
- For example, if calculating ${\frac {13}{2}}\times {\frac {4}{9}}$ , you would multiply the denominators: $2\times 9=18$ . Putting together your numerator and denominator, your answer becomes ${\frac {52}{18}}$ .
5
Simplify your answer, if possible. To simplify, or reduce, a fraction, you need to find the greatest factor (besides 1) that is common to the numerator and the denominator. Then, divide the numerator and denominator by that factor. For more information on this process, read Reduce Fractions.
- For example, $52$ and $18$ are both divisible by $2$ .
  $52\div 2=26$
  $18\div 2=9$
  So, ${\frac {52}{18}}={\frac {26}{9}}$

Part 3

Part 3 of 3:

Converting Improper Fractions Back Into Mixed Numbers

1
Divide the numerator by the denominator. If there is no remainder, then your answer is a whole number rather than a mixed number, and you need not do anything further. Likely, though, you will have a remainder. Set this aside for now. The quotient you found when Dividing the numerator by the denominator will be the whole number of your mixed number.^{[13]
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Research source}
- For example, $26\div 9=2$ with a remainder of $8$ . Thus, the whole number of your mixed number will be 2.
2
Turn the remainder into the numerator of your fraction. Place this numerator over the original denominator. This will give you the fraction of your mixed number.
- For example, if your original denominator is $9$ and your remainder is $8$ , the fraction of your mixed number is ${\frac {8}{9}}$ .
3
Combine the whole number and the fraction. This gives you the final answer to your original division problem.

Community Q&A

Question

How do I calculate 2 5/7 divided by 3 1/2?

Donagan

Top Answerer

Change both mixed numbers to improper fractions. Invert the second improper fraction, then multiply it by the first fraction to get the answer. Reduce it if possible, and convert it to a mixed number if desired.
Question

To find the reciprocal, I flip the second fraction. What if I have fraction divided by fraction divided by fraction? Would I flip the last fraction, or the second one?

Donagan

Top Answerer

Flip both the second and third fractions.
Question

Why do you flip the fractions and then multiply in Part 2, step 2?

Community Answer

Any fraction multiplied by its reciprocal is equal to 1. For example, 5/10 x 10/5 = 50/50 = 1. When you divide by a fraction, you want to cancel the second fraction by changing it to a 1. So, to do this, you have to multiply it by its reciprocal (or flip it). But, if you're going to multiply the second fraction by its reciprocal, you also have to multiply the other number by its reciprocal. So when you "flip and multiply" you are really cancelling out the second fraction, which leaves you with just multiplying the first fraction by the second fraction's reciprocal.