Divide a Unit Fraction by a Whole Number | Grade 5 Math | LevelKite

You know that a fraction can come from sharing. Split 1 whole into 4 equal shares, and each share is $\frac{1}{4}$ of the whole: $1 \div 4 = \frac{1}{4}$ .

A fraction with 1 as its top number, such as $\frac{1}{4}$ , is called a unit fraction. It names one equal part of a whole.

What if you split one of those shares again? The division $\frac{1}{4} \div 2$ asks: if you split $\frac{1}{4}$ of a whole into 2 equal shares, how much of the whole is one share? Here you divide a unit fraction by a whole number, a number such as 2, 3 or 4.

A model is a picture that shows the numbers. Let's start with a model of $\frac{1}{4}$ . The bar that comes next is one whole.

One of the 4 equal parts is shaded, so the shaded part is

\frac{1}{4}

of the whole.

Now divide the shaded fourth by 2. Cut it into 2 equal parts. Each part is one share. One share is half of the fourth.

But what fraction of the whole is one share? A unit fraction names one part only when all the parts of the whole are equal. So cut every fourth into 2 equal parts, not just the shaded one.

The next model shows one whole after that cut. It has 4 columns side by side. Each column is one fourth. The line across cuts each fourth into 2 equal parts. One share is shaded.

Each of the 4 fourths is cut into 2 equal parts, so the whole has 8 equal parts and the 1 shaded part is

\frac{1}{8}

of the whole.

Count the equal parts in the whole. There are 4 fourths with 2 parts in each. That makes $4 \times 2 = 8$ equal parts.

One share is 1 of those 8 equal parts. So one share is $\frac{1}{8}$ of the whole.

Now write what the model shows, using numbers:

$\frac{1}{4} \div 2 = \frac{1}{8}$

The answer to a division is called the quotient. Here the quotient is $\frac{1}{8}$ . Notice two things about it.

First, it is a unit fraction. One share is exactly 1 of the equal parts of the whole, so its top number is 1.

Second, it is smaller than $\frac{1}{4}$ . One share is only part of the fourth, so it must be smaller than the fourth. The bottom numbers show this too. The bottom number of a fraction is its denominator. It tells how many equal parts make the whole. The whole went from 4 equal parts to 8 equal parts. More equal parts in the same whole means each part is smaller.

This happens every time you divide a unit fraction by a whole number that is 2 or more.

Worked example

Divide with a model

Use a model to find $\frac{1}{2} \div 3$ .

Show the unit fraction first. Cut one whole into 2 equal parts. Each part is one half of the whole.
Divide by 3. Cut one half into 3 equal parts, because dividing by 3 makes 3 equal shares. Cut the other half the same way, so that all the parts of the whole are equal.
Count the equal parts in the whole. There are 2 halves with 3 parts in each, so multiply.
$2 \times 3 = 6$
Name one share. It is 1 of the 6 equal parts of the whole, so it is one sixth. The model after this example shows it: 2 columns for the halves, 3 equal parts in each column, and 1 part shaded.
$\frac{1}{2} \div 3 = \frac{1}{6}$

Answer

One half divided by 3 is one sixth: $\frac{1}{2} \div 3 = \frac{1}{6}$ . One share is $\frac{1}{6}$ of the whole, which is smaller than $\frac{1}{2}$ .

Each of the 2 halves is cut into 3 equal parts, so the whole has 6 equal parts and the 1 shaded part is

\frac{1}{6}

of the whole.

Compare the two models. Look at where each new denominator comes from.

Division	The cut in the model	Equal parts in the whole	Quotient
$\frac{1}{4} \div 2$	each of the 4 fourths into 2 equal parts	$4 \times 2 = 8$	$\frac{1}{8}$
$\frac{1}{2} \div 3$	each of the 2 halves into 3 equal parts	$2 \times 3 = 6$	$\frac{1}{6}$

Key concept

Cut one part, then count the whole

To divide a unit fraction by a whole number, cut the one part into that many equal parts. One of the new parts is the quotient.

Every part of the whole gets the same cut, so that all the parts are equal. To count the equal parts in the whole, multiply the denominator you started with by the whole number. The answer to a multiplication is called a product. That product is the denominator of the quotient. The top number stays 1, because one share is one of those equal parts.

\frac{1}{4} \div 2 = \frac{1}{4 \times 2} = \frac{1}{8}

Worked example

Divide without drawing

Find $\frac{1}{5} \div 4$ .

Think about the cuts. The denominator 5 tells you the whole has 5 equal parts, called fifths. To divide one fifth by 4, cut it into 4 equal parts. Cut every other fifth the same way, so all the parts are equal.
Count the equal parts in the whole. There are 5 fifths with 4 parts in each, so multiply the denominator by the whole number.
$5 \times 4 = 20$
One share is 1 of those 20 equal parts, so the quotient is 1 over 20.
$\frac{1}{5} \div 4 = \frac{1}{20}$

Answer

One fifth divided by 4 is one twentieth: $\frac{1}{5} \div 4 = \frac{1}{20}$ . The quotient is a smaller unit fraction than $\frac{1}{5}$ , as it should be.

Worked example

Share what is left

There is $\frac{1}{3}$ of a pan of brownies left. Four friends share it equally. What fraction of the whole pan does each friend get?

Write the division. The piece that is left is one third of the pan, and 4 friends share it equally, so divide one third by 4.
$\frac{1}{3} \div 4$
Think of every third of the pan cut into 4 equal parts. Multiply to count the equal parts in the whole pan.
$3 \times 4 = 12$
One friend gets 1 of those 12 equal parts, so the share is one twelfth of the pan.
$\frac{1}{3} \div 4 = \frac{1}{12}$

Answer

Each friend gets $\frac{1}{12}$ of the whole pan.

Remember: dividing a unit fraction by a whole number means cutting one part of the whole into that many equal parts. The whole then has more equal parts, so one share is a smaller unit fraction. Its denominator is the denominator you started with times the whole number, as in $\frac{1}{4} \div 2 = \frac{1}{8}$ , where $4 \times 2 = 8$ .

Next, you will turn the question around. You will divide a whole number by a unit fraction, and you will see why that quotient is greater than the whole number.