Lesson 3: Dividing by a Fraction

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. For example, to divide by $\frac{2}{3}$, you multiply by its reciprocal, $\frac{3}{2}$.

First, convert any mixed numbers to improper fractions. Then, apply the rule:

Examples

• How many $\frac{3}{4}$-foot lengths of rope can be cut from a 6-foot rope? We calculate $6 \div \frac{3}{4} = 6 \times \frac{4}{3} = \frac{24}{3} = 8$ lengths.

Property

If $a$, $b$, $c$, and $d$ are numbers where $b \neq 0$, $c \neq 0$, and $d \neq 0$, then

To divide fractions, multiply the first fraction by the reciprocal of the second.

Examples

• To divide $\frac{1}{3} \div \frac{1}{9}$, we multiply $\frac{1}{3}$ by the reciprocal of $\frac{1}{9}$, which is $\frac{9}{1}$. So, $\frac{1}{3} \cdot \frac{9}{1} = \frac{9}{3} = 3$.
• Let's divide $\frac{5}{6} \div \frac{10}{3}$. We keep $\frac{5}{6}$, change to multiplication, and flip $\frac{10}{3}$ to $\frac{3}{10}$. This gives $\frac{5}{6} \cdot \frac{3}{10} = \frac{15}{60}$, which simplifies to $\frac{1}{4}$.
• For $-\frac{2}{5} \div \frac{4}{15}$, we calculate $-\frac{2}{5} \cdot \frac{15}{4}$. The result is negative. $\frac{2 \cdot 15}{5 \cdot 4} = \frac{30}{20}$. This simplifies to $-\frac{3}{2}$.

Explanation

To divide fractions, use the 'Keep, Change, Flip' method. Keep the first fraction the same, change the division sign to multiplication, and flip the second fraction to its reciprocal. Then, simply multiply the fractions as usual.