Lesson 3: Multiply Unit and Non-unit Fractions

Property

To multiply two fractions, we can find a "fraction of a fraction." An area model visualizes this by representing the first fraction with vertical divisions and the second fraction with horizontal divisions. The product is the area where the shaded parts of both fractions overlap.

Examples

Property

To multiply a unit fraction by a non-unit fraction, multiply the numerators together and the denominators together. For fractions $\frac{1}{b}$ and $\frac{c}{d}$:

Examples

• $\frac{1}{3} \times \frac{2}{5} = \frac{1 \times 2}{3 \times 5} = \frac{2}{15}$
• $\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$

Explanation

This skill extends the area model to multiply a unit fraction by a non-unit fraction. For example, to find $\frac{1}{3}$ of $\frac{2}{5}$, you first model $\frac{2}{5}$ and then take $\frac{1}{3}$ of that shaded area. The resulting product is found by multiplying the numerators to find the number of shaded parts and multiplying the denominators to find the total number of parts in the whole. This bridges the gap between multiplying two unit fractions and multiplying two non-unit fractions.