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Dividing Fractions

by Ron Kurtus (revised 18 February 2009)

A fraction is a part of the whole or the quotient of two whole numbers. Dividing a fraction by another fraction involves some trickery. You invert or form the reciprocal of the second fraction and then multiply the two fractions. When you are working with a fraction and whole number, the whole number must be converted to a fraction. Likewise, mixed numbers must be changed into improper fractions before the division operation.

Questions you may have include:

This lesson will answer those questions.



Dividing one fraction by another

The way to divide one fraction by another fraction is to simply invert or flip the fraction you are dividing by (the second fraction) and then multiply the two fractions together.

Example

The following example demonstrates this method:

1/4 ÷ 3/7 =

Flip over 3/7 or create its reciprocal and then change sign to multiplication:

1/4 × 7/3 =

Multiply top numbers (numerators) and bottom numbers (denominators):

(1 × 7)/(4 × 3) =

7/12

Another example

Another example is:

12/13 ÷ 1/4 =

Invert 1/4 and change sign to multiplication:

12/13 × 4/1 =

Multiply numerators and denominators:

(12 × 4)/(13 × 1) =

48/13

Reduce improper fraction:

48 ÷ 13 = 3 9/13

Dividing fraction and whole number

Sometimes you encounter a division problem with a fraction and a whole number.

Divide fraction by whole number

When you divide by a whole number, you can put the number in the form of an improper fraction and then flip it over. You take the whole number and show it as divided by 1, such as 7 = 7/1.

An example of dividing a fraction by a whole number is:

3/4 ÷ 5 =

Put 5 in fraction format:

3/4 ÷ 5/1 =

Flip over 5/1 and change to multiplication:

3/4 × 1/5 =

Multiply numerators and denominators:

(3 × 1)/(4 × 5) =

3/20

Divide a whole number by a fraction

You can also divide a whole number by a fraction in the same manner. Change the whole number into a fraction over 1 and then flip the fraction you are dividing by.

17 ÷ 2/3 =

Put 17 in fraction format:

17/1 ÷ 2/3 =

Flip over 2/3 and change to multiplication:

17/1 × 3/2 =

Multiply numerators and denominators:

(17 × 3)/(1 × 2) =

51/2

Reduce improper fraction:

51 ÷ 2 = 25 1/2

Dividing with mixed numbers

When you divide using mixed numbers, you need to change the mixed number into a fraction first.

Divide a mixed number by a fraction

5 1/3 ÷ 2/7 =

Change mixed number to a fraction:

(5 × 3 + 1)/3 = 16/3

Thus:

16/3 ÷ 2/7 =

Flip over 2/7:

16/3 × 7/2 =

Multiply numerators and denominators:

(16 × 7)/(3 × 2) =

112/6 =

Reduce fraction by dividing top and bottom by 2:

56/3 =

Reduce improper fraction:

18 2/3

Divide a fraction by a mixed number

2/3 ÷ 1 2/7 =

(Change mixed number to a fraction)

1 2/7 = 7/7 + 2/7 = 9/7

2/3 ÷ 9/7 =

(Flip over 9/7)

2/3 × 7/9 =

(Multiply numerators and denominators)

(2 × 7)/(3 × 9)

= 14/27

Divide a mixed number by a mixed number

2 1/2 ÷ 3 1/3 =

Change 2 1/2 to a fraction:

5/2 ÷ 3 1/3 =

Change 3 1/3 to a fraction:

5/2 ÷ 10/3 =

Flip over 10/3 and multiply:

5/2 × 3/10 =

(5 × 3)/(2 × 10) =

15/20

Divide numerator and denominate by 5 to reduce to a proper fraction:

(15 ÷ 5)/(20 ÷ 5) = 3/4

Summary

Dividing a fraction by another fraction involves inverting the second fraction that you are dividing by and then multiplying the two fractions. When you are working with a fraction and whole number, the whole number must be changed to an improper fraction. Likewise, mixed numbers must be changed into improper fractions before the division operation.


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