Dividing Fractions

What You’ll Learn

In this lesson you’ll learn the “keep-change-flip” rule for dividing fractions, how to handle whole numbers and mixed numbers, and why it works.

The Concept

Dividing fractions finds how many times one fraction fits into another. Rule: “Keep the first fraction, change $\div$ to $\times$ , flip the second fraction (reciprocal).”

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Example: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1 \frac{7}{8}$

Why? Dividing is “how many of the second fit into the first,” and multiplying by the reciprocal gives that count.

Whole number $\div$ fraction: $6 \div \frac{2}{3} = \frac{6}{1} \times \frac{3}{2} = \frac{18}{2} = 9$
Fraction $\div$ whole: $\frac{2}{3} \div 4 = \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$

Mixed numbers: Convert to improper first.

Worked Examples

Divide $\frac{5}{6} \div \frac{3}{8}$

Keep $\frac{5}{6}$ , change $\div$ to $\times$ , flip $\frac{3}{8}$ to $\frac{8}{3}$
$\frac{5}{6} \times \frac{8}{3} = \frac{40}{18} = \frac{20}{9} = 2 \frac{2}{9}$

Now: $3 \frac{1}{2} \div 1 \frac{1}{4}$

Convert to improper: $\frac{7}{2} \div \frac{5}{4}$
$\frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = \frac{14}{5} = 2 \frac{4}{5}$

Real-World Applications

Dividing fractions solves “how many” questions: How many $\frac{1}{3}$ -cup servings in $2 \frac{1}{2}$ cups? ( $2 \frac{1}{2} \div \frac{1}{3} = 7 \frac{1}{2}$ servings). Or fabric: 5 yards $\div$ $\frac{3}{4}$ yard per piece $= 6 \frac{2}{3}$ pieces. Or rates: speed = distance $\div$ time (e.g., 60 miles $\div$ $1 \frac{1}{2}$ hours $= 40$ mph).

Quiz

What is 1/2 ÷ 1/4?

A.2
B.1/8
C.1/2
D.4

Divide 3/5 ÷ 2/3.

A.9/15
B.9/10
C.1/2
D.6/5

A tank holds 4 1/2 gallons. How many 3/4-gallon containers can it fill?

A.6
B.5
C.6 1/2
D.5 1/2

Divide 2 2/3 ÷ 1 1/3.

A.2
B.3
C.1 1/2
D.2 1/2

What is 5 ÷ 1/3?

A.15
B.5/3
C.1/15
D.3/5