Lesson 23: Multiplying and Dividing Mixed Numbers

New Concept

To multiply or divide mixed numbers, we first write each mixed number as an improper fraction. Here is how we convert:

What’s next

Soon, we’ll walk through worked examples of both multiplication and division, and then apply this skill to solve a construction-based word problem.

Property

To handle multiplication or division, first convert mixed numbers into improper fractions. To do this, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Examples

• $$3\frac{1}{2} = \frac{2 \times 3 + 1}{2} = \frac{7}{2}$$
• $$5\frac{2}{3} = \frac{3 \times 5 + 2}{3} = \frac{17}{3}$$
• $$4\frac{3}{5} = \frac{5 \times 4 + 3}{5} = \frac{23}{5}$$

Explanation

Think of it as counting slices! A mixed number like $3\frac{1}{2}$ means three whole pizzas and one extra slice. To make it one big fraction, you need to count all the slices. The shortcut is to multiply the whole number by the bottom number (denominator) and then add the top number (numerator) to get your new total.

Property

First, rewrite each mixed number as an improper fraction. Then, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Finally, simplify the resulting fraction if possible.

Examples

• $$2\frac{1}{2} \times 1\frac{1}{5} \rightarrow \frac{5}{2} \times \frac{6}{5} = \frac{30}{10} = 3$$
• $$3\frac{1}{3} \times 1\frac{1}{2} \rightarrow \frac{10}{3} \times \frac{3}{2} = \frac{\cancel{10}^5}{\cancel{3}_1} \times \frac{\cancel{3}^1}{\cancel{2}_1} = \frac{5}{1} = 5$$

Explanation

Multiplying mixed numbers is a two-step dance! First, you have to change your fancy mixed numbers into their simpler improper fraction outfits. Once they're dressed for the party, it's easy: top times top, and bottom times bottom. Then just simplify your answer so it looks sharp, maybe by changing it back into a mixed number.

Property

First, convert all mixed numbers to improper fractions. Then, instead of dividing, multiply the first fraction by the reciprocal (the flipped version) of the second fraction. Simplify the result.

Examples

• $$4\frac{1}{2} \div 1\frac{1}{2} \rightarrow \frac{9}{2} \div \frac{3}{2} = \frac{9}{2} \times \frac{2}{3} = \frac{18}{6} = 3$$
• $$3\frac{1}{3} \div 1\frac{2}{3} \rightarrow \frac{10}{3} \div \frac{5}{3} = \frac{\cancel{10}^2}{3} \times \frac{3}{\cancel{5}_1} = \frac{2 \times 3}{3 \times 1} = 2$$

Explanation

Dividing mixed numbers has a cool secret: don't divide, multiply! First, turn your mixed numbers into improper fractions. Then, take the second fraction, flip it upside down (this is its reciprocal), and multiply instead. It’s the classic 'keep, change, flip' move that turns a tricky division problem into a much simpler multiplication one.