Divide Whole Numbers by Unit Fractions | Grade 5 Math | LevelKite

You know that $6 \div 2$ can ask: how many 2s fit in 6? Three 2s fit, so $6 \div 2 = 3$ . The answer to a division is called the quotient.

In this lesson you divide a whole number, such as 2 or 5, by a unit fraction. A unit fraction names one of the equal parts of a whole, such as $\frac{1}{3}$ or $\frac{1}{4}$ . Its top number is always 1.

The division asks the same kind of question. $2 \div \frac{1}{3}$ asks: how many thirds fit in 2 wholes? You have already used models to divide a unit fraction by a whole number. A model helps here too.

2

wholes, each cut into

3

equal parts, make

6

thirds:

2 \div \frac{1}{3} = 6

.

Look at the model above. Here is how it matches $2 \div \frac{1}{3}$ :

The 2 bars are the 2 wholes you start with.
Each bar is cut into 3 equal parts, so each part is $\frac{1}{3}$ of a whole.
Every part is shaded, because you count every one. There are 3 parts in the first bar and 3 in the second, so there are 6 parts in all.

So 6 thirds fit in 2 wholes. As a number sentence, that is $2 \div \frac{1}{3} = 6$ . The quotient is 6.

Worked example

Use a model to divide

Use a model to find $3 \div \frac{1}{4}$ .

Say what the division asks. Dividing by one fourth means counting fourths. So the question is: how many fourths fit in 3 wholes?
Draw 3 bars to show the 3 wholes. Cut each bar into 4 equal parts, because 4 fourths make 1 whole. Now every part is one fourth. The model after this example shows the drawing.
$1 \text{ whole} = 4 \text{ fourths}$
Count the parts in all 3 bars. Each bar has 4 parts, so count 4, then 4 more, then 4 more.
$4 + 4 + 4 = 12$
All 12 parts are fourths, so 12 fourths fit in 3 wholes. Write this as a division sentence.
$3 \div \frac{1}{4} = 12$

Answer

There are $12$ fourths in $3$ wholes, so $3 \div \frac{1}{4} = 12$ .

3

wholes, each cut into

4

equal parts, make

12

fourths:

3 \div \frac{1}{4} = 12

.

Did you notice something about the two quotients? In $2 \div \frac{1}{3} = 6$ , the quotient 6 is greater than 2. In $3 \div \frac{1}{4} = 12$ , the quotient 12 is greater than 3.

The models show why. A unit fraction such as $\frac{1}{3}$ or $\frac{1}{4}$ is smaller than 1 whole, so more than one of them fits in each whole. Each bar is just 1 whole, but it adds 3 or 4 parts to your count. So there are more parts than wholes. That is why the quotient is greater than the whole number you started with.

The models also show a faster way to count. Look again at the model for $3 \div \frac{1}{4}$ . It has 3 bars with 4 parts in each bar. That is 3 equal groups of 4, so you can multiply instead of counting one by one: $3 \times 4 = 12$ .

Where do the 3 and the 4 come from? The 3 is the whole number. The 4 is the denominator of $\frac{1}{4}$ . The denominator is the bottom number of a fraction. It tells how many equal parts make one whole, so it also tells how many parts are in each bar.

The first model works the same way. For $2 \div \frac{1}{3}$ , there are 2 bars with 3 parts in each bar, and $2 \times 3 = 6$ .

Key concept

Count the parts in every whole

Dividing a whole number by a unit fraction asks how many parts of that size fit in the wholes. The denominator tells you how many parts fit in one whole, and every whole holds that many. So the quotient is the whole number times the denominator. It is their product, the answer you get when you multiply them. For example, $3 \div \frac{1}{4} = 3 \times 4 = 12$ .

\text{number of wholes} \times \text{parts in each whole} = \text{parts in all}

Worked example

Multiply instead of counting one by one

Find $5 \div \frac{1}{6}$ without drawing every part.

Start with what the division asks: how many sixths fit in 5 wholes? The denominator is 6, so each whole holds 6 sixths.
$1 \text{ whole} = 6 \text{ sixths}$
There are 5 wholes with 6 sixths in each. That is 5 equal groups of 6, so multiply the whole number by the denominator.
$5 \times 6 = 30$
The 30 parts are sixths, so 30 sixths fit in 5 wholes. Write this as a division sentence.
$5 \div \frac{1}{6} = 30$

Answer

There are $30$ sixths in $5$ wholes, so $5 \div \frac{1}{6} = 30$ .

2

wholes, each cut into

2

halves, make

4

halves:

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

.

Worked example

Divide and explain the size

Find $12 \div \frac{1}{5}$ . Then explain why the quotient is greater than $12$ .

The denominator is 5, so each whole holds 5 fifths. Multiply the 12 wholes by 5 to count all the fifths.
$12 \div \frac{1}{5} = 12 \times 5 = 60$
Explain the size. One fifth is smaller than 1 whole, so each of the 12 wholes holds more than one fifth. That makes the number of fifths greater than the number of wholes.
$60 > 12$

Answer

There are $60$ fifths in $12$ wholes, so $12 \div \frac{1}{5} = 60$ . The quotient is greater than $12$ because every whole holds $5$ fifths, not just one.

Remember: dividing a whole number by a unit fraction means counting how many parts of that size fit in the wholes. Cut each whole into equal parts and count them all. Every whole holds as many parts as the denominator says, so you can multiply the whole number by the denominator. The quotient is greater than the whole number because more than one part fits in each whole.

Next, you will check divisions like these with multiplication, and you will learn to find and fix mistakes.