Lesson 22: Multiplying and Dividing Fractions

New Concept

To multiply fractions we multiply the numerators to find the numerator of the product, and we multiply the denominators to find the denominator of the product.

What’s next

Next, you’ll see this rule visualized with area models, then learn how its inverse—division—works using reciprocals in worked examples.

Property

To multiply fractions, multiply the numerators to find the numerator of the product, and multiply the denominators to find the denominator of the product.

Examples

• To solve $\frac{1}{4} \cdot \frac{3}{5}$, we multiply the numerators and denominators: $\frac{1 \cdot 3}{4 \cdot 5} = \frac{3}{20}$.
• What is $\frac{2}{5}$ of $\frac{5}{8}$? We can cancel before multiplying: $\frac{\stackrel{1}{\cancel{2}}}{\underset{1}{\cancel{5}}} \cdot \frac{\stackrel{1}{\cancel{5}}}{\underset{4}{\cancel{8}}} = \frac{1}{4}$.
• Simplify $\frac{2}{3} \cdot \frac{4}{5}$: $\frac{2 \cdot 4}{3 \cdot 5} = \frac{8}{15}$.

Explanation

Ever wondered what a 'half of a quarter' really is? Multiplying fractions lets you find a part of a part! Just as a pint is half of a quart (which is a quarter of a gallon), you multiply the top numbers and the bottom numbers together to find the final, smaller fraction. It’s a simple way to combine parts.

Property

If the product of two fractions is 1, the fractions are reciprocals. Another name for a reciprocal is a multiplicative inverse.

Examples

• The reciprocal of $\frac{7}{9}$ is $\frac{9}{7}$.
• The multiplicative inverse of 5 (which is $\frac{5}{1}$) is $\frac{1}{5}$.
• The number of $\frac{2}{5}$s in 1 is the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$.

Explanation

Think of a reciprocal as a fraction's 'upside-down' twin! When you multiply a fraction by its reciprocal, they magically cancel out to equal 1. This 'flipping' action is the secret key to making division with fractions super easy. Mastering this move turns confusing division problems into simple multiplications you already know how to solve.

Property

To divide by a fraction, multiply by its reciprocal (multiplicative inverse).

Examples

• Simplify $\frac{1}{3} \div \frac{1}{9}$. We multiply by the reciprocal: $\frac{1}{3} \times \frac{9}{1} = \frac{9}{3} = 3$.
• To solve $\frac{2}{5} \div \frac{3}{4}$, we rewrite it as a multiplication problem: $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$.
• Simplify $3 \div \frac{5}{6}$. We treat 3 as $\frac{3}{1}$ and multiply by the reciprocal: $\frac{3}{1} \times \frac{6}{5} = \frac{18}{5} = 3\frac{3}{5}$.

Explanation

Dividing fractions is like asking, 'How many little pieces fit into my bigger piece?' Instead of doing tricky division, we use a cool shortcut: keep the first fraction, change division to multiplication, and flip the second fraction (use its reciprocal). This 'keep-change-flip' move transforms a division problem into a straightforward multiplication problem you already know!