Lesson 68: Dividing Mixed Numbers

New Concept

To divide mixed numbers, first convert them into improper fractions. Then, multiply the first fraction by the reciprocal of the second fraction.

What’s next

This is the foundational method. Next, you'll tackle worked examples and challenge problems to solidify your understanding of dividing mixed numbers.

Property

To divide mixed numbers, you must complete two steps. First, rewrite all mixed numbers or whole numbers as improper fractions. Then, to perform the division, multiply the first fraction by the reciprocal of the second fraction.

Examples

$3\frac{1}{3} \div 1\frac{1}{2} = \frac{10}{3} \div \frac{3}{2} = \frac{10}{3} \times \frac{2}{3} = \frac{20}{9} = 2\frac{2}{9}$
$2\frac{1}{2} \div 1\frac{2}{3} = \frac{5}{2} \div \frac{5}{3} = \frac{5}{2} \times \frac{3}{5} = \frac{15}{10} = 1\frac{1}{2}$
$8 \div 1\frac{1}{3} = \frac{8}{1} \div \frac{4}{3} = \frac{8}{1} \times \frac{3}{4} = \frac{24}{4} = 6$

Explanation

Think of this as a two-part mission! First, you have to get your mixed numbers into their 'improper fraction' disguises. Then, instead of a direct fight (division), you bring in the second number's secret agent twin (its reciprocal) and multiply. It's a clever switcheroo that makes solving the problem much simpler!

Property

Dividing by a number is the same as multiplying by its reciprocal. For any numbers $a$ and $b$ (where $b \ne 0$), $a \div b = a \times \frac{1}{b}$.

Examples

Shawna dividing $2\frac{2}{3}$ cups for 4 plants ($2\frac{2}{3} \div 4$) is the same as finding $\frac{1}{4}$ of the food ($2\frac{2}{3} \times \frac{1}{4}$).
To solve $1\frac{3}{5} \div 4$, you can instead calculate $1\frac{3}{5} \times \frac{1}{4} = \frac{8}{5} \times \frac{1}{4} = \frac{2}{5}$.
Compare: $4\frac{1}{2} \div 3$ is equal to $\frac{1}{3} \times 4\frac{1}{2}$, since dividing by 3 is the same as multiplying by $\frac{1}{3}$.

Explanation

Why is dividing by 4 the same as multiplying by $\frac{1}{4}$? It’s two ways of asking the same thing! Dividing asks 'how many groups of 4 can I make?' while multiplying by $\frac{1}{4}$ asks 'what is one-fourth of my stuff?' Both questions split your total into four equal shares, leading to the exact same answer.