Lesson 7: Division of Whole Numbers by Unit Fractions

Property

Dividing a whole number, $a$, by a unit fraction, $\frac{1}{b}$, is a way of asking: "How many groups of size $\frac{1}{b}$ are in $a$?"
This can be modeled visually to find the total number of fractional parts.

Examples

Property

We noticed in the previous visual models that dividing by a unit fraction ($\frac{1}{b}$) produces the same result as multiplying by the denominator ($b$).
To solve $a \div \frac{1}{b}$, you can ask: "$a$ is $\frac{1}{b}$ of what number?". The shortcut is to multiply the whole number by the denominator:

Examples

Property

To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the fraction. This is because you are finding how many fractional parts fit into the whole number.

Examples

• How many $\frac{1}{4}$-cup servings are in 3 cups of sugar?

• A ribbon is 5 meters long. How many $\frac{1}{3}$-meter pieces can be cut from it?

Explanation

Dividing a whole number by a unit fraction asks the question, "How many of these fractional pieces fit into the whole amount?" For example, $2 \div \frac{1}{4}$ is asking how many quarter-pieces fit into 2 wholes. Since there are 4 quarters in 1 whole, there must be $2 \times 4 = 8$ quarters in 2 wholes. This concept is the inverse of dividing a fraction by a whole number.