Lesson 26: Multiplying and Dividing Mixed Numbers

New Concept

To multiply or divide mixed numbers, first rewrite them as improper fractions. Then, you can multiply or divide the improper fractions as indicated.

What’s next

This card covers the foundational method. Next, you'll see worked examples for finding area, dividing portions, and even squaring mixed numbers.

Property

To multiply mixed numbers, first convert them into improper fractions.

Examples

$3\frac{2}{3} \times 1\frac{1}{2} = \frac{11}{3} \times \frac{3}{2} = \frac{11}{2} = 5\frac{1}{2}$
$3 \times 2\frac{1}{2} = \frac{3}{1} \times \frac{5}{2} = \frac{15}{2} = 7\frac{1}{2}$

Explanation

Don't multiply the whole numbers and fractions separately! That's a trap. First, transform each mixed number into a top-heavy improper fraction. This combines everything into a simple format. Now, you just multiply the numerators and multiply the denominators to find your answer. It is the best method.

Property

Convert the mixed number to an improper fraction, then multiply it by itself.

Examples

$(1\frac{1}{2})^2 = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4} = 2\frac{1}{4}$
$(2\frac{1}{2})^2 = \frac{5}{2} \times \frac{5}{2} = \frac{25}{4} = 6\frac{1}{4}$

Explanation

Squaring means multiplying a number by itself, but there’s a crucial first step for mixed numbers. Never square the whole and fraction parts separately! First, convert the mixed number into an improper fraction. Then it is easy—just multiply that fraction by itself to get the correct squared value every time.

Property

First, convert all mixed numbers to improper fractions. Then, multiply the first fraction by the reciprocal of the second.

Examples

$1\frac{1}{3} \div 2\frac{1}{2} = \frac{4}{3} \div \frac{5}{2} = \frac{4}{3} \times \frac{2}{5} = \frac{8}{15}$
$2\frac{1}{2} \div 3\frac{1}{3} = \frac{5}{2} \div \frac{10}{3} = \frac{5}{2} \times \frac{3}{10} = \frac{15}{20} = \frac{3}{4}$

Explanation

Turn division into easy multiplication! Step one: change all your mixed numbers into improper fractions. Step two: use the 'Keep-Change-Flip' strategy. You keep the first fraction, change the sign to multiplication, and flip the second fraction upside down. Now you just have to multiply across and simplify your answer.