Chapter 2 · Fraction review

Multiply Fractions by Whole Numbers and by Fractions

Learn to multiply a fraction by a whole number and to multiply two fractions, and see why you multiply the numerators and the denominators.

14 min read

In the last lesson you multiplied fractions with area models. Now you will get the same answers from the numbers alone, with no drawing.

Two words first. A product is the result of multiplying. A factor is a number that you multiply.

Start with a whole number, such as $3$ , times a fraction. The product $3 \times \frac{2}{7}$ means $3$ groups of $\frac{2}{7}$ . You can find it by adding $\frac{2}{7}$ three times.

3 \times \frac{2}{7} = \frac{2}{7} + \frac{2}{7} + \frac{2}{7} = \frac{2 + 2 + 2}{7} = \frac{3 \times 2}{7} = \frac{6}{7}

Look at what changed and what stayed the same.

The denominator is the bottom number. It tells how many equal pieces make one whole, so it names the size of each piece. Every piece here is a seventh, so the denominator stays $7$ .

The numerator is the top number. It counts the pieces. Three groups of $2$ pieces make $3 \times 2 = 6$ pieces.

This gives a rule for any whole number and any fraction: multiply the whole number by the numerator and keep the denominator.

The fraction can also come first. $\frac{2}{3} \times 6$ means $\frac{2}{3}$ of $6$ . To find it from the meaning, split $6$ into $3$ equal parts. Each part is $2$ . Take $2$ of those parts: $2 + 2 = 4$ .

The rule gives the same answer: $\frac{2 \times 6}{3} = \frac{12}{3} = 4$ . The last step works because $3$ thirds make one whole, so $12$ thirds make $12 \div 3 = 4$ wholes.

So the rule is the same in either order: the whole number multiplies the numerator, and the denominator stays.

Worked example

A whole number times a fraction

What is $5 \times \frac{3}{8}$ ?

Read the product as 5 groups of 3 eighths. Every piece is an eighth, so the denominator stays 8.
Multiply the whole number by the numerator, because 5 groups of 3 pieces make 5 times 3 pieces, which is 15 pieces.
$5 \times \frac{3}{8} = \frac{5 \times 3}{8} = \frac{15}{8}$
Look for a simpler form. A fraction is in simplest form when no whole number greater than 1 divides both its numerator and its denominator. That is true for 15 and 8, so this fraction is already in simplest form. A numerator larger than the denominator is fine: 8 eighths make 1 whole, so 15 eighths is 1 whole and 7 more eighths.
$\frac{15}{8} = \frac{8}{8} + \frac{7}{8} = 1 + \frac{7}{8}$
Check the size. The number you started with, 3 eighths, is positive, and you multiplied it by 5, which is greater than 1. So the product must be greater than 3 eighths. It is, because 15 eighths is more than 3 eighths.
$\frac{15}{8} > \frac{3}{8}$

Answer

The product $5 \times \frac{3}{8}$ is $\frac{15}{8}$ .

You can use a size check like the one at the end of that example for any product. It is only true under the conditions listed here. Choose one of the two numbers as the starting quantity. The other number is the factor you multiply it by.

When the starting quantity is positive (greater than $0$ ):

a factor greater than $1$ makes the product greater than the starting quantity;
a factor between $0$ and $1$ makes the product less than the starting quantity;
a factor of exactly $1$ leaves the starting quantity unchanged.

When the starting quantity is $0$ , the product is $0$ for every factor. Zero stays zero.

This check is for multiplying only. Division has its own pattern: dividing a positive quantity by a positive number below $1$ can increase it.

The table shows each multiplying case with real numbers.

The size check with real numbers

Starting quantity	Factor	Product	Compared with the start
$\frac{3}{8}$	$5$ , greater than $1$	$\frac{5 \times 3}{8} = \frac{15}{8}$	Greater: $\frac{15}{8} > \frac{3}{8}$
$12$	$\frac{3}{2}$ , greater than $1$ because $\frac{2}{2} = 1$	$\frac{12 \times 3}{2} = \frac{36}{2} = 18$	Greater: $18 > 12$
$12$	$\frac{2}{3}$ , between $0$ and $1$	$\frac{12 \times 2}{3} = \frac{24}{3} = 8$	Less: $8 < 12$
$12$	$\frac{3}{3}$ , which is exactly $1$	$\frac{12 \times 3}{3} = \frac{36}{3} = 12$	Unchanged: $12 = 12$
$0$	$\frac{2}{3}$	$\frac{0 \times 2}{3} = \frac{0}{3} = 0$	Zero stays zero: $0 = 0$

A common mistake is to multiply both the numerator and the denominator by the whole number. For $5 \times \frac{3}{8}$ , the mistake gives $\frac{5 \times 3}{5 \times 8} = \frac{15}{40}$ .

Here is why it fails. Multiplying the denominator by $5$ cuts every eighth into $5$ smaller pieces. You get $5$ times as many pieces, but each piece is only one fifth as large. So the amount has not changed: $\frac{15}{40} = \frac{15 \div 5}{40 \div 5} = \frac{3}{8}$ .

But $\frac{3}{8}$ is positive and $5$ is a factor greater than $1$ , so the product must be greater than $\frac{3}{8}$ . Five groups means five times as many pieces of the same size. Multiply only the numerator: $5 \times \frac{3}{8} = \frac{15}{8}$ .

Now multiply two fractions. The product $\frac{2}{3} \times \frac{4}{5}$ means $\frac{2}{3}$ of $\frac{4}{5}$ : a fraction of a fraction.

Think of the area model from the last lesson. A rectangle stands for one whole. Cut it into $5$ equal columns and shade $4$ of them to show $\frac{4}{5}$ . Then cut it into $3$ equal rows and keep $2$ rows of the shaded columns.

The cuts make $3 \times 5 = 15$ equal parts in the whole.
You keep $2$ rows of $4$ parts, which is $2 \times 4 = 8$ parts.

So the product is $8$ of the $15$ equal parts.

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

Key concept

Multiply the numerators, multiply the denominators

To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. The new denominator counts the equal parts in one whole. The new numerator counts the parts you keep.

In the formula, $\frac{a}{b}$ and $\frac{c}{d}$ stand for any two fractions. The numerators are $a$ and $c$ . The denominators are $b$ and $d$ , and a denominator is never $0$ .

A whole number follows the same rule. Write it as a fraction with denominator $1$ . For example, $5 = \frac{5}{1}$ , because $5 \div 1 = 5$ . Then $\frac{5}{1} \times \frac{3}{8} = \frac{5 \times 3}{1 \times 8} = \frac{15}{8}$ , the same product you found before.

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \quad \text{for } b \neq 0 \text{ and } d \neq 0

Worked example

Two fractions

What is $\frac{3}{4} \times \frac{5}{7}$ ?

Multiply the numerators. This counts the parts you keep.
$3 \times 5 = 15$
Multiply the denominators. This counts the equal parts in one whole.
$4 \times 7 = 28$
Write the product as a fraction. No whole number greater than 1 divides both 15 and 28, so it is already in simplest form.
$\frac{3}{4} \times \frac{5}{7} = \frac{3 \times 5}{4 \times 7} = \frac{15}{28}$
Check the size. Take 5 sevenths as the starting quantity. It is positive, and the factor 3 fourths is between 0 and 1, so the product must be less than 5 sevenths. To compare, rename 5 sevenths in twenty-eighths. Multiplying its numerator and denominator by 4 cuts each piece into 4 smaller pieces, so the amount stays the same.
$\frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28}, \quad \frac{15}{28} < \frac{20}{28}$

Answer

The product $\frac{3}{4} \times \frac{5}{7}$ is $\frac{15}{28}$ .

Worked example

A fraction of an amount in a recipe

A recipe uses $\frac{3}{4}$ cup of milk. You make $\frac{2}{3}$ of the recipe. How many cups of milk do you need?

You need 2 thirds of 3 fourths of a cup. A fraction of an amount means that fraction times the amount, so multiply.
$\frac{2}{3} \times \frac{3}{4}$
Multiply the numerators and multiply the denominators, as the rule for two fractions says.
$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12}$
Simplify. The number 6 divides both 6 and 12. Dividing the numerator and the denominator by the same nonzero number regroups the pieces into larger ones, so the amount stays the same.
$\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}$
Check the size. The starting quantity, 3 fourths of a cup, is positive, and the factor 2 thirds is between 0 and 1. So you need less than 3 fourths of a cup. One half equals 2 fourths, which is less than 3 fourths.
$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} < \frac{3}{4}$

Answer

You need $\frac{1}{2}$ cup of milk.

Takeaway. Multiplying fractions is counting parts. Multiply the numerators to count the parts you keep. Multiply the denominators to count the equal parts in one whole. A whole number such as $5$ is $\frac{5}{1}$ , so the same rule covers a whole number times a fraction: the numerator is multiplied and the denominator stays. Then simplify, and use the size check with its conditions to see whether your product is reasonable.

Next, you will estimate products of fractions and mixed numbers, and you will multiply mixed numbers.

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Estimate and Multiply Mixed Numbers

14 min read

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