Lesson 2: Dividing Fractions

Property

The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$, where $a \neq 0$ and $b \neq 0$. A number and its reciprocal have a product of 1. To find the reciprocal of a fraction, we invert the fraction. Reciprocals must have the same sign. The number zero does not have a reciprocal.

Examples

• The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$. Check: $\frac{5}{8} \cdot \frac{8}{5} = \frac{40}{40} = 1$.
• The reciprocal of $9$ is $\frac{1}{9}$. First, write $9$ as $\frac{9}{1}$, then invert it. Check: $9 \cdot (\frac{1}{9}) = 1$.
• The reciprocal of $\frac{1}{4}$ is $4$. Check: $\frac{1}{4} \cdot (4) = 1$.

Explanation

Finding a reciprocal is like flipping a fraction upside down. The numerator becomes the denominator and the denominator becomes the numerator. When you multiply any number by its reciprocal, the result is always 1. Keep the sign the same!

Property

To divide a number by a fraction, multiply the number by the reciprocal of the fraction.

The reciprocal of a fraction is found by interchanging the numerator and the denominator. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.

Examples

• How many $\frac{1}{4}$-cup scoops of sugar are in a 2-cup bag? This is $2 \div \frac{1}{4}$. We calculate $2 \times \frac{4}{1} = 8$. There are 8 scoops.

Property

To divide a whole number by a non-unit fraction, we can use a two-step process or simply multiply by the reciprocal.
The reciprocal of a fraction $\frac{b}{c}$ is $\frac{c}{b}$ (flipping the numerator and denominator).

Examples

Property

To divide a fraction by a whole number, you apply the same rule: multiply the fraction by the reciprocal of the whole number.
Since the reciprocal of a whole number $c$ is $\frac{1}{c}$, this operation makes the fractional parts smaller.

Examples