How to Divide Mixed Numbers

A mixed number is a combination of an integer and a fraction. We can divide two mixed numbers together, however, you need to convert them to non-conforming fractions first. After the mixed number is converted, you can do division as you would with fractions.

Table of Contents

Convert mixed numbers to improper fractions

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Multiply the denominator in the fraction by the whole number of the mixed number. ^{[1] X Research Source} Perform this step with both mixed numbers and then set the results aside. This is just part of the new numerator.

Example: calculate $6$ $first$ $2$ $\div$ $2$ $first$ $4$ ${displaystyle 6{frac {1}{2}}div 2{frac {1}{4}}}$ ${displaystyle 6{frac {1}{2}}div 2{frac {1}{4}}}$ . First you need to get $6$ $\times$ $2$ $=$ $twelfth$ ${displaystyle 6times 2=12}$ ${displaystyle 6times 2=12}$ and $2$ $\times$ $4$ $=$ $8$ ${displaystyle 2times 4=8}$ ${displaystyle 2times 4=8}$ .

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Add the last result with the numerator. ^{[2] X Research Source} Perform this step with both mixed numbers. The sum you find will be the numerator of the wrong fraction.

Continuing the example, we have $twelfth$ $+$ $first$ $=$ $13$ ${displaystyle 12+1=13}$ ${displaystyle 12+1=13}$ and $8$ $+$ $first$ $=$ $9$ ${displaystyle 8+1=9}$ ${displaystyle 8+1=9}$ .

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Put the sum you just found above the original denominator. ^{[3] X Research Source} Complete this step with both fractions and check to make sure you are using the correct denominator. These will be two mismatched fractions that we will use to do the division.

In the example above, $6$ $first$ $2$ ${displaystyle 6{frac {1}{2}}}$ ${displaystyle 6{frac {1}{2}}}$ will become $13$ $2$ ${displaystyle {frac {13}{2}}}$ ${displaystyle {frac {13}{2}}}$ and $2$ $first$ $4$ ${displaystyle 2{frac {1}{4}}}$ ${displaystyle 2{frac {1}{4}}}$ converted to $9$ $4$ ${displaystyle {frac {9}{4}}}$ ${displaystyle {frac {9}{4}}}$ .

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Convert integers to fractions. If either number is an integer, you need to convert it to a fraction by dividing the integer by 1. At this point, the integer becomes the numerator. ^{[4] X Research Sources}

For example: $3$ $=$ $3$ $first$ ${displaystyle 3={frac {3}{1}}}$ ${displaystyle 3={frac {3}{1}}}$ .

Divide fractions that don’t fit

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Rewrite the problem with the converted fraction. Use the improper fractions you converted to complete the division in Part 1.

The problem will become: $13$ $2$ $\div$ $9$ $4$ ${displaystyle {frac {13}{2}}div {frac {9}{4}}}$ ${displaystyle {frac {13}{2}}div {frac {9}{4}}}$ .

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Inverse the second fraction. ^{[5] X Research Source} To find the reciprocal of a fraction, you need to “turn it upside down” so that the numerator becomes the denominator and vice versa. ^{[6] X Research Source} Then change the problem to multiplication. ^{[7] X Research Sources}

In the above example, the inverse of $9$ $4$ ${displaystyle {frac {9}{4}}}$ ${displaystyle {frac {9}{4}}}$ To be $4$ $9$ ${displaystyle {frac {4}{9}}}$ ${displaystyle {frac {4}{9}}}$ . So, $13$ $2$ $\div$ $9$ $4$ ${displaystyle {frac {13}{2}}div {frac {9}{4}}}$ ${displaystyle {frac {13}{2}}div {frac {9}{4}}}$ will become $13$ $2$ $\times$ $4$ $9$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$

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Multiply the numerators together. To proceed, you multiply the two numerators together as if they were two whole numbers. This product will be the numerator of the answer. ^{[8] X Research Sources}

Continuing the example, to calculate $13$ $2$ $\times$ $4$ $9$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$ , you need to multiply the numerators: $13$ $\times$ $4$ $=$ $52$ ${displaystyle 13times 4=52}$ ${displaystyle 13times 4=52}$ .

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Multiply the denominators together. To proceed, you multiply the two denominators together as if they were two whole numbers. This product will be the denominator of the answer. ^{[9] X Research Source}

In the above example, to calculate $13$ $2$ $\times$ $4$ $9$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$ ${displaystyle {frac {13}{2}}times {frac {4}{9}}}$ Then you need to multiply the two denominators: $2$ $\times$ $9$ $=$ $18$ ${displaystyle 2times 9=18}$ ${displaystyle 2times 9=18}$ . Then combine the numerator and denominator to get the result: $52$ $18$ ${displaystyle {frac {52}{18}}}$ ${displaystyle {frac {52}{18}}}$ .

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Simplify results if necessary. To reduce or simplify a fraction, we need to find the greatest common divisor (other than 1), which is the number that both the numerator and the denominator are divisible by. Then take the numerator and denominator and divide by that number. To learn more about how, see Simplify fractions.

In the example, we see $52$ ${displaystyle 52}$ ${displaystyle 52}$ and $18$ ${displaystyle 18}$ ${displaystyle 18}$ are divisible by $2$ ${displaystyle 2}$ ${displaystyle 2}$ .
$52$ $\div$ $2$ $=$ $26$ ${displaystyle 52div 2=26}$ ${displaystyle 52div 2=26}$
$18$ $\div$ $2$ $=$ $9$ ${displaystyle 18div 2=9}$ $18div 2=9$
So, $52$ $18$ $=$ $26$ $9$ ${displaystyle {frac {52}{18}}={frac {26}{9}}}$ ${displaystyle {frac {52}{18}}={frac {26}{9}}}$

Convert the wrong fraction to a mixed number

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Divide the numerator by the denominator. If the division has no remainder, we will have an integer instead of a mixed number and stop here. But if the division doesn’t run out and there is a remainder, put the remainder aside. The quotient of the numerator and denominator you just found will be the integer in the mixed number. ^{[10] X Research Source}

Continuing the example in the previous section: we have $26$ $\div$ $9$ $=$ $2$ ${displaystyle 26div 9=2}$ ${displaystyle 26div 9=2}$ residual $8$ ${displaystyle 8}$ ${displaystyle 8}$ . Therefore, the integer of this mixed number will be 2.

Steps

Convert mixed numbers to improper fractions

Divide fractions that don’t fit

Convert the wrong fraction to a mixed number

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