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Divide Fractions

$\require{cancel}$Once you know how to multiply fractions, it's VERY easy to divide fractions.  You just find the reciprocal of the second fraction (flip it, so the the denominator becomes the numerator and the numerator becomes the denominator), and then multiply the fractions.  It's really that easy!

Why does this algorithm work?

Division is multiplication's inverse operation.  They are opposites.  Fractions themselves are division problems, the number on the top divided by the number on the bottom.  So, when you flip the divisor fraction and multiply, you get a division problem.  Let's look at an example:

$$\eqalign{7\div 2 = 3\dfrac{1}{2} \qquad \qquad 7 \times \dfrac{1}{2}=\dfrac {7}{2}=7 \div 2=3 \dfrac{1}{2}}$$

As you can see, dividing 7 by 2 is the same as multiplying 7 by $\dfrac{1}{2}$ and $\dfrac{1}{2}$ is the reciprocal of $\dfrac{2}{1}$ or $2$.

Although it's easy to see how this works with whole numbers (multiplying any number times the reciprocal of a whole number, is the same as dividing by that whole number), the process works in exactly the same way for all fractions.  Flip the divisor fraction and multiply. After you find the reciprocal of the divisor, all other multiplication rules apply, so feel free to cross-cancel!


$$\eqalign{\dfrac{5}{6}\div\dfrac{1}{12}=&\\\dfrac{5}{6}\times\dfrac{12}{1}=&\qquad&&\text{Replace the division fraction with its reciprocal and multiply}\\\dfrac{5}{1}\times\dfrac{2}{1}=&\dfrac{10}{1}&&\text{Multiply numerator times numerator and denominator times denominator}\\=&10 &&\text {Simplify to whole number}}$$

Dividing fractions is easy! Just find the reciprocal of the second (divisor) fraction and then multiply.  For more on multiplying fractions see lessons Multiply Fractions (basic), Multiply Fractions (advanced), Multiply Fractions with Mixed Numbers.

Alternate Method:

A common memory trick that teachers use for fraction division is "Keep, change, flip."

When you approach a fraction division problem:

$\dfrac{1}{2} \div \dfrac{1}{3}=$

You "keep" the first fraction as it is, change the $\div$ to a $\times$, and then "flip" the second fraction to its reciprocal.

$\dfrac{1}{2} \times \dfrac{3}{1}=$

Then, you just multiply the fractions.

Practice Problems:

  • Fraction Division (No Mixed Numbers, No Cross-Cancel)

    Find the quotient. Simplify all answers completely. Change improper fractions to mixed numbers. 

    1. $\dfrac{1}{3}$ $\div$ $\dfrac{2}{5}=$

    2. $\dfrac{2}{9}$ $\div$ $\dfrac{4}{5}=$

    3. $\dfrac{2}{3}$ $\div$ $\dfrac{7}{8}=$

    4. $\dfrac{1}{7}$ $\div$ $\dfrac{1}{3}=$

    5. $\dfrac{1}{8}$ $\div$ $\dfrac{7}{10}=$

    6. $\dfrac{5}{18}$ $\div$ $\dfrac{9}{15}=$

    7. $\dfrac{4}{15}$ $\div$ $\dfrac{1}{8}=$

    8. $\dfrac{6}{7}$ $\div$ $\dfrac{1}{2}=$

    9. $\dfrac{2}{6}$ $\div$ $\dfrac{9}{16}=$

    10. $\dfrac{7}{8}$ $\div$ $\dfrac{4}{5}=$

    11. $\dfrac{2}{7}$ $\div$ $\dfrac{1}{12}=$

    12. $\dfrac{6}{7}$ $\div$ $\dfrac{17}{18}=$