Lesson 8: Divide Unit Fractions by Whole Numbers

Property

Dividing a unit fraction $\frac{1}{a}$ by a whole number $b$ can be represented by partitioning the fractional piece into $b$ equal smaller pieces.

Examples

• To solve $\frac{1}{2} \div 3$, you can draw a rectangle, shade in $\frac{1}{2}$ of it, and then divide that shaded portion into 3 equal parts. Each new part represents $\frac{1}{6}$ of the whole rectangle.
• To solve $\frac{1}{4} \div 2$, you can draw a circle, shade in $\frac{1}{4}$ of it, and then divide that shaded wedge into 2 equal parts. The new, smaller wedge represents $\frac{1}{8}$ of the whole circle.

Explanation

Visual models help you understand what it means to divide a fraction by a whole number. First, you represent the unit fraction with a diagram, such as shading part of a shape. Then, you divide that shaded area by the whole number by splitting it into that many equal sections. The size of one of these new, smaller sections compared to the whole shape is the answer to the division problem.

Property

To divide a unit fraction by a whole number, you multiply the unit fraction by the reciprocal of the whole number. The reciprocal of a whole number $c$ is $\frac{1}{c}$.

Examples

• $\frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$
• $\frac{1}{5} \div 4 = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20}$
• $\frac{1}{8} \div 2 = \frac{1}{8} \times \frac{1}{2} = \frac{1}{16}$

Explanation

Dividing a unit fraction by a whole number means splitting an already small piece into even smaller, equal parts. For example, dividing $\frac{1}{2}$ by $3$ is like cutting half a pizza into $3$ equal slices. The procedure for this is to change the division problem into a multiplication problem by using the reciprocal of the whole number. This method connects division and multiplication, showing they are inverse operations.