Multiply a Fraction by a Whole Number: Count the Parts | Grade 5 Math | LevelKite

You already know how to count things: 1 apple, 2 apples, 3 apples. In this lesson you will count parts of a whole in the same way. That counting shows you how to multiply a fraction by a whole number such as 3 or 4.

Take the fraction $\frac{5}{4}$ . The bottom number is the denominator. It tells you that one whole is cut into 4 equal parts, so each part is a fourth. The top number is the numerator. It tells you how many of those parts you have: 5.

A unit fraction names just one of the equal parts of a whole. Its numerator is 1, as in $\frac{1}{4}$ or $\frac{1}{8}$ .

You can count unit fractions out loud: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths. So $\frac{5}{4}$ is 5 parts, and each part is $\frac{1}{4}$ .

Each bar is one whole cut into 4 equal parts. Five parts are shaded, so the shaded amount is

\frac{5}{4}

.

Look at the two bars. Each bar is one whole cut into 4 equal parts, so each part is $\frac{1}{4}$ . Five parts are shaded. In words, that is five fourths.

You already add fractions. When the parts are the same size, you count them. So 5 copies of $\frac{1}{4}$ add up to $\frac{5}{4}$ .

Adding 5 copies of a number is what multiplying that number by 5 means. A multiple of a number is what you get when you multiply it by a whole number. So $\frac{5}{4}$ is a multiple of the unit fraction $\frac{1}{4}$ . The figure, the words and the number sentence all show the same amount:

\frac{5}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 5 \times \frac{1}{4}

Worked example

Write a fraction as a multiple of a unit fraction

Write $\frac{3}{8}$ as a whole number times a unit fraction.

Find the size of the parts. The denominator is 8, so one whole is cut into 8 equal parts and each part is an eighth. That is the unit fraction.
$\frac{1}{8}$
Find how many parts there are. The numerator is 3, so there are 3 eighths. Three copies of one eighth is 3 times one eighth.
$\frac{3}{8} = 3 \times \frac{1}{8}$
Check by adding the 3 copies. The parts are all eighths, so you count them: 1 eighth, 2 eighths, 3 eighths.
$\frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$

Answer

As a multiple of a unit fraction, $\frac{3}{8} = 3 \times \frac{1}{8}$ .

So far, each numerator has counted unit fractions: $\frac{5}{4}$ is 5 fourths and $\frac{3}{8}$ is 3 eighths. This works for a fraction greater than 1 and for a fraction less than 1.

Now multiply a whole number by a fraction that is not a unit fraction: $3 \times \frac{2}{5}$ .

With whole numbers, $3 \times 2$ means 3 equal groups of 2. It works the same way here. $3 \times \frac{2}{5}$ means 3 equal groups of $\frac{2}{5}$ . Each group is 2 fifths, because $\frac{2}{5} = 2 \times \frac{1}{5}$ .

The figure is a model of this multiplication, a picture that shows it. The 3 groups are shaded on bars cut into fifths, 2 parts for each group.

Two bars, each cut into 5 equal parts. Six parts are shaded: 3 groups of 2 fifths, so

3 \times \frac{2}{5} = \frac{6}{5}

.

Worked example

Use a model to multiply

Use the bars shown just before this example to find $3 \times \frac{2}{5}$ . Name the total as a fraction and as a mixed number. A mixed number is a whole number and a fraction written together, such as $2\frac{1}{2}$ for 2 wholes and 1 half.

Read the multiplication as equal groups: 3 groups of 2 fifths. Each bar is one whole cut into 5 equal parts, so each part is one fifth and each group is 2 parts.
$3 \times \frac{2}{5} = \frac{2}{5} + \frac{2}{5} + \frac{2}{5}$
Count the shaded parts in twos, one group at a time: 2, 4, 6. Three groups of 2 parts make 6 shaded parts. Every part is still one fifth, so the total is 6 fifths.
$3 \times \frac{2}{5} = \frac{6}{5}$
The total is more than one whole, so you can also name it as a mixed number. Five of the shaded fifths fill one whole bar. That leaves 1 fifth shaded in the other bar, so the total is 1 whole and 1 fifth.
$\frac{6}{5} = \frac{5}{5} + \frac{1}{5} = 1 + \frac{1}{5} = 1\frac{1}{5}$

Answer

Three groups of $\frac{2}{5}$ make 6 fifths, so $3 \times \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5}$ .

Key concept

Multiply the numerator, keep the denominator

To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.

The model shows why the denominator stays. In $3 \times \frac{2}{5}$ , one group is 2 shaded parts and three groups are 6 shaded parts. So the whole number changes how many parts there are. It does not change the size of a part: every bar stays cut into fifths. The numerator counts the parts, so it is multiplied. The denominator names the size of the parts, so it stays 5.

3 \times \frac{2}{5} = \frac{3 \times 2}{5} = \frac{6}{5}

Worked example

Multiply with the rule, then check

Find $4 \times \frac{2}{3}$ . Give the answer as a fraction and as a mixed number.

Multiply the numerator by the whole number. There are 4 groups of 2 thirds, and 4 groups of 2 parts is 8 parts. Keep the denominator 3, because the parts are still thirds.
$4 \times \frac{2}{3} = \frac{4 \times 2}{3} = \frac{8}{3}$
Write 8 thirds as a mixed number. Three thirds make 1 whole, so 6 thirds make 2 wholes. That uses 6 of the 8 thirds and leaves 2 thirds.
$\frac{8}{3} = \frac{6}{3} + \frac{2}{3} = 2 + \frac{2}{3} = 2\frac{2}{3}$
Check with the model shown right after this example. It has 3 bars, each cut into 3 equal parts, with 8 parts shaded. Counting the shaded parts in twos gives 4 groups. Two bars are full and 2 thirds of the last bar are shaded, which matches the answer.

Answer

So $4 \times \frac{2}{3} = \frac{8}{3}$ , which is $2\frac{2}{3}$ as a mixed number.

Three bars, each cut into 3 equal parts. Eight parts are shaded:

4 \times \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3}

.

The big idea: a fraction is a count of equal parts. $\frac{5}{4}$ is 5 parts of size $\frac{1}{4}$ , so $\frac{5}{4} = 5 \times \frac{1}{4}$ .

When you multiply a fraction by a whole number, you make that many equal groups. The whole number multiplies how many parts you have. It does not change the size of a part. So you multiply the numerator and keep the denominator.

Next, you will find a fraction of a whole number by splitting the number into equal parts and taking some of them. You will also use both ideas to solve word problems.