Math Core

A recipe needs $\dfrac{2}{3}$ cup of milk, but you only want to make half of it. How much milk is half of $\dfrac{2}{3}$ cup? Questions like this, "a fraction of an amount," are answered by multiplying fractions.

A whole number times a fraction

$3 \times \dfrac{2}{5}$ means 3 groups of $\dfrac{2}{5}$ :

\frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{6}{5}

You get 6 fifths. Only the number of pieces changed, not the size of the pieces, so multiply the whole number by the numerator and keep the denominator: $3 \times \dfrac{2}{5} = \dfrac{3 \times 2}{5} = \dfrac{6}{5} = 1\dfrac{1}{5}$ .

A fraction of a whole number

$\dfrac{2}{3} \times 12$ means $\dfrac{2}{3}$ of 12. Split 12 into 3 equal groups of 4, then take 2 of the groups: $2 \times 4 = 8$ .

You get the same answer by multiplying the numerator by 12 and dividing by the denominator: $\dfrac{2 \times 12}{3} = \dfrac{24}{3} = 8$ .

A fraction times a fraction

What is $\dfrac{2}{3}$ of $\dfrac{4}{5}$ ? Start with a square that stands for 1 whole. Shade $\dfrac{4}{5}$ of it with columns. Then shade $\dfrac{2}{3}$ of it with rows. The part shaded twice is $\dfrac{2}{3}$ of $\dfrac{4}{5}$ .

The whole square has 5 × 3 = 15 small pieces. The darkest part, shaded twice, has 4 × 2 = 8 of them.

The whole is cut into $5 \times 3 = 15$ pieces, and the double-shaded part has $4 \times 2 = 8$ pieces. So

\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

Multiplying fractions

Multiply the numerators, and multiply the denominators:

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

To use this with a whole number, write the whole number over 1. For example, $6 = \dfrac{6}{1}$ . To use it with a mixed number, first change the mixed number to a fraction greater than 1.

Worked example: A whole number times a fraction

Find $4 \times \dfrac{2}{3}$ .

4 \times \frac{2}{3} = \frac{4}{1} \times \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3}

Worked example: A fraction of a whole number

Mia has 20 stickers. She gives away $\dfrac{3}{4}$ of them. How many does she give away?

\frac{3}{4} \times 20 = \frac{3 \times 20}{4} = \frac{60}{4} = 15

She gives away 15 stickers. Check: $\dfrac{1}{4}$ of 20 is 5, so $\dfrac{3}{4}$ of 20 is $3 \times 5 = 15$ .

Worked example: Half of a fraction

Back to the recipe: what is $\dfrac{1}{2}$ of $\dfrac{2}{3}$ cup?

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

You need $\dfrac{1}{3}$ cup of milk.

Worked example: A mixed number

Find $1\dfrac{1}{2} \times \dfrac{4}{5}$ .

Change $1\dfrac{1}{2}$ to halves: $1\dfrac{1}{2} = \dfrac{3}{2}$ . Then

\frac{3}{2} \times \frac{4}{5} = \frac{12}{10} = \frac{6}{5} = 1\frac{1}{5}

Common mistake

When you multiply a whole number by a fraction, don't multiply the denominator too. $3 \times \dfrac{2}{5}$ is not $\dfrac{6}{15}$ . Write the 3 as $\dfrac{3}{1}$ , and you will see the denominator is multiplied by 1.

Tip

You can simplify before you multiply. In $\dfrac{3}{4} \times \dfrac{8}{9}$ , the 8 and 4 share a factor of 4, and the 3 and 9 share a factor of 3. You get $\dfrac{1}{1} \times \dfrac{2}{3} = \dfrac{2}{3}$ with smaller numbers.

Practice

Practice 1

Find $5 \times \dfrac{1}{4}$ .

Type a number.

Practice 2

Find $\dfrac{2}{5} \times 10$ .

Type a number.

Practice 3

Find $\dfrac{1}{3} \times \dfrac{1}{4}$ .

Type a number.

Practice 4

Find $\dfrac{3}{4} \times \dfrac{2}{5}$ .

Type a number.

Practice 5

A class has 18 students, and $\dfrac{2}{3}$ of them walk to school. How many students walk to school?

Type a number.

Practice 6

Find $\dfrac{5}{6} \times \dfrac{3}{10}$ .

Type a number.

Practice 7

Find $2\dfrac{1}{2} \times \dfrac{4}{5}$ .

Type a number.

Practice 8

One batch of muffins uses $\dfrac{3}{4}$ cup of sugar. How many cups of sugar are in $2\dfrac{1}{2}$ batches?

Type a number.