Math Core

You know that $4 \times 3$ means 4 groups of 3. So what could $4 \times \dfrac{2}{3}$ mean? The same thing: 4 groups of $\dfrac{2}{3}$ . Multiplying a fraction by a whole number is just repeated adding of fractions.

Start with unit fractions

A pizza is cut into 8 slices. Five friends each eat 1 slice, which is $\dfrac{1}{8}$ of the pizza. How much pizza did they eat?

\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{5}{8}

That's 5 groups of $\dfrac{1}{8}$ , so $5 \times \dfrac{1}{8} = \dfrac{5}{8}$ . In fact, any fraction is a whole number times a unit fraction: $\dfrac{5}{8} = 5 \times \dfrac{1}{8}$ .

Groups of any fraction

Now try $4 \times \dfrac{2}{3}$ . That's 4 jumps of $\dfrac{2}{3}$ on a number line:

4 jumps of 2 thirds. Each +2 means 2 thirds. You land on 8/3.

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

Look at the numbers: 4 groups of 2 thirds is $4 \times 2 = 8$ thirds. The pieces are still thirds.

Multiplying a fraction by a whole number

Multiply the whole number by the numerator. Keep the denominator the same.

n \times \frac{a}{b} = \frac{n \times a}{b}

This works because $n \times \dfrac{a}{b}$ means $n$ groups of $a$ pieces, and each piece is still $\dfrac{1}{b}$ .

Worked example: Using a unit fraction

Find $6 \times \dfrac{1}{4}$ .

6 groups of 1 fourth is 6 fourths:

6 \times \frac{1}{4} = \frac{6}{4} = 1\frac{2}{4}.

Worked example: Any fraction

Find $3 \times \dfrac{3}{5}$ .

Multiply the whole number by the numerator: $3 \times 3 = 9$ . Keep fifths.

3 \times \frac{3}{5} = \frac{9}{5} = 1\frac{4}{5}

Worked example: A word problem

Each lap around the school track is $\dfrac{3}{8}$ of a mile. Kai runs 5 laps. How far does he run? Between which two whole numbers of miles is that?

5 \times \frac{3}{8} = \frac{15}{8}

$15 \div 8 = 1$ with a remainder of 7, so $\dfrac{15}{8} = 1\dfrac{7}{8}$ miles. That's between 1 and 2 miles, very close to 2.

Common mistake

Don't multiply the denominator too. $4 \times \dfrac{2}{3}$ is not $\dfrac{8}{12}$ . The whole number tells how many groups there are. It doesn't change the size of the pieces, so the denominator stays 3.

Tip

Quick check: if the fraction is less than 1, then $n$ times it must be less than $n$ . For example, $5 \times \dfrac{3}{8}$ has to be less than 5.

Practice

Practice 1

Find $3 \times \dfrac{1}{7}$ .

Type your answer like 2/3.

Type a number.

Practice 2

Which multiplication matches this number line?

Each +3 means 3 fifths.

Practice 3

Find $5 \times \dfrac{2}{6}$ .

Type your answer like 2/3.

Type a number.

Practice 4

Fill in the blank: $\dfrac{7}{10} = ? \times \dfrac{1}{10}$ .

Type a number.

Practice 5

Which is equal to $4 \times \dfrac{3}{4}$ ?

Practice 6

A recipe uses $\dfrac{3}{4}$ of a cup of milk. Rosa makes the recipe 3 times. How many cups of milk does she need?

Type your answer like 1 2/3.

Type a number.

Practice 7

Each bag of trail mix holds $\dfrac{2}{3}$ of a pound. Mr. Diaz buys 8 bags. Between which two whole numbers of pounds is the total?

Practice 8

Jada hikes $\dfrac{5}{12}$ of a mile each day. After how many days will she have hiked $\dfrac{20}{12}$ of a mile?

Type a number.