How to divide fractions
Before reading about how to divide fractions, please read the primer on fractions.
Let's kick off with this simple example of dividing fractions
1/3 ÷ 1/2
Change the division sign (÷) to a multiplication sign (x) and 'flip' the second fraction
1/3 x 2/1
It may help to imagine a pancake marked on one side with the fraction 1/2 - 'flip' the pancake to reveal the fraction 2/1 on the other side!
Now follow the rules for multiplying fractions -
multiply the numerators
1 x 2 = 2
multiply the denominators
3 x 1 = 3
The resulting fraction is 2/3.
So
1/3 ÷ 1/2 = 1/3 x 2/1 = (1 x 2) / (3 x 1) = 2/3
i.e. one third divided by a half equals two thirds.
Next consider
1/7 ÷ 1/2 ÷ 2/3
Change the division signs (÷) to multiplication signs (x) and 'flip' the second and third fractions
1/7 x 2/1 x 3/2
Now multiply the numerators
1 x 2 x 3 = 6
and multiply the denominators
7 x 1 x 2 = 14
Simplify the resulting fraction 6/14 to the equivalent fraction 3/7 by dividing numerator and denominator by factor 2
Bringing everything together
1/7 ÷ 1/2 ÷ 2/3
=
1/7 x 2/1 x 3/2
=
(1 x 2 x 3) / (7 x 1 x 2)
=
6/14
=
3/7
i.e. one seventh divided by a half divided by two thirds equals three sevenths.
Have a go at the following
3/4 ÷ 3/4 ÷ 1/2 ÷ 1/2
5/3 ÷ 2/2 ÷ 1/2 ÷ 2/3 ÷ 1/2
Don't worry about how many fractions are involved or the improper fractions (5/3 and 2/2, numerator greater than or equal to denominator). Apply the same technique as shown in the previous examples of how to divide fractions.
To recap, change the division signs (÷) to multiplication signs (x) and 'flip' all the fractions except the first fraction. Then follow the rules for multiplying fractions - multiply the numerators and multiply the denominators.
The answers/workings out are as follows
3/4 ÷ 3/4 ÷ 1/2 ÷ 1/2
=
3/4 x 4/3 x 2/1 x 2/1
=
(3 x 4 x 2 x 2) / (4 x 3 x 1 x 1)
=
48 / 12
(48 divided by 12)
=
4
i.e. three quarters divided by three quarters divided by a half divided by a half equals 4.
5/3 ÷ 2/2 ÷ 1/2 ÷ 2/3 ÷ 1/2
=
5/3 x 2/2 x 2/1 x 3/2 x 2/1
=
(5 x 2 x 2 x 3 x 2) / (3 x 2 x 1 x 2 x 1)
=
120/12
(120 divided by 12)
=
10
i.e. five thirds divided by two halves divided by a half divided by two thirds divided by a half equals 10.
Feeling more confident about how to divide fractions?
Using the technique you've just learned, go ahead and divide fractions with confidence - regardless of the number of fractions or whether they're proper or improper!
For more on fractions click here:
Mixed fractions (mixed numbers)
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