## Dividing Fractions with Whole Numbers

A fraction is a part of a whole. The given pizza is cut into 5 equal slices and 3 of slices are left. That is, 3 out of 5 slices of pizza are there. The fraction shown is ^{3}⁄_{5}.

Now, if we divide this three-fifths of the pizza into 3 equal parts, each part will have one part out of the 5 parts as shown.

That is 35÷3= 1⁄_{5}.

Dividing ^{3}⁄_{5} by 3, will give us one-third of ^{3}⁄_{5}.

That is ^{3}⁄_{5 }x ^{1}⁄_{3 }= ^{1}⁄_{5}.

We can also verify this as ^{1}⁄_{5 }x 3= ^{3}⁄_{5}.

Consider dividing the fraction ^{4}⁄_{6} by 2.

The portion shaded in pink is divided equally into two parts – shaded in green and the one in blue respectively. The part in green is ^{3}⁄_{6} of the rectangle, and so is the part shaded in blue.

That is 462=26.

We can verify this using multiplication as ^{2}⁄_{6 }x 2 = ^{4}⁄_{6}.

Here again, on dividing ^{4}⁄_{6} by 2, we precisely find the one-half of ^{4}⁄_{6}.

That is ^{4}⁄_{6 }x ^{1}⁄_{2}= ^{2}⁄_{6}.

In both the examples, in the procedure, the division symbol is replaced with multiplication, and its multiplicative inverse or reciprocal replaces the divisor.

The rule is, to divide a fraction by a whole number, multiply the given fraction by the reciprocal of whole numbers.

Example: Find ^{1}⁄_{4}÷3.

The reciprocal of 3 is ^{1}⁄_{3}.

^{1}⁄_{4}÷3 = ^{1}⁄_{4 }x ^{1}⁄_{3 }= ^{1}⁄_{12}

Conceptually this can be shown as:

Example: If 5 out of 12 pieces of an apple pie were shared among 3 people, what fraction of apple pie does each person gets?

We know that ^{5}⁄_{12} of the apple pie is equally shared between 3 people.

So, we have to find ^{5}⁄_{12 }÷3.

The reciprocal of 3 is ^{1}⁄_{3}.

^{5}⁄_{12 }÷ 3 = ^{5}⁄_{12 }x ^{1}⁄_{3 }= ^{5}⁄_{36}

Therefore, each person gets ^{5}⁄_{36} of the apple pie.

Fun Facts:

What if the divisor is a fraction? The rule remains the same!

^{3}⁄_{4 }÷ ^{1}⁄_{4} = ^{3}⁄_{4 }x 4_{ }= 3

How many ¼ s are there in a ¾?

There are 3 one-fourth in a three-fourth!

- If the dividend is a mixed fraction, first convert the mixed fraction into an improper fraction and apply the rule.

Example: 5^{1}⁄_{3 }÷ 8

First, convert 5^{1}⁄_{3} into an improper fraction.

5^{1}⁄_{3 }= ^{(5x3)+1}⁄_{3 }= ^{16}⁄_{3}

Now, 5^{1}⁄_{3 }÷ 8 = ^{16}⁄_{3} ÷ 8 = ^{16}⁄_{3} x ^{1}⁄_{8} = ^{2}⁄_{3} .