Dividing Fractions

Dividing fractions can feel confusing at first, but it relies on one simple idea: dividing by a fraction is the same as multiplying by its reciprocal. Once you learn the “Keep-Change-Flip” method, dividing fractions becomes just as straightforward as multiplying them.

What Is a Reciprocal?

The reciprocal of a fraction is what you get when you flip the numerator and denominator.

$\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}$

A number multiplied by its reciprocal always equals 1:

$\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$

The Keep-Change-Flip Method

To divide fractions:

1. Keep the first fraction as it is
2. Change the division sign to multiplication
3. Flip the second fraction (use its reciprocal)

Then multiply using the standard fraction multiplication rules.

Formula:

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

Example 1: Divide $\frac{3}{4} \div \frac{2}{5}$

Keep $\frac{3}{4}$, Change $\div$ to $\times$, Flip $\frac{2}{5}$ to $\frac{5}{2}$:

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

Answer: $1\frac{7}{8}$

Example 2: Divide $\frac{5}{6} \div \frac{5}{12}$

Keep-Change-Flip:

$\frac{5}{6} \div \frac{5}{12} = \frac{5}{6} \times \frac{12}{5}$

Cross-cancel: 5 and 5 cancel to 1, and 12 and 6 share a factor of 6:

$\frac{\cancel{5}^{1}}{\cancel{6}_{1}} \times \frac{\cancel{12}^{2}}{\cancel{5}_{1}} = \frac{1 \times 2}{1 \times 1} = 2$

Answer: $2$

Example 3: Divide $\frac{7}{8} \div \frac{3}{4}$

$\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3}$

Cross-cancel: 4 and 8 share a factor of 4:

$\frac{7}{\cancel{8}_{2}} \times \frac{\cancel{4}^{1}}{3} = \frac{7 \times 1}{2 \times 3} = \frac{7}{6} = 1\frac{1}{6}$

Answer: $1\frac{1}{6}$

Dividing a Whole Number by a Fraction

Write the whole number as a fraction over 1, then apply Keep-Change-Flip.

Example 4: Divide $6 \div \frac{3}{4}$

$\frac{6}{1} \div \frac{3}{4} = \frac{6}{1} \times \frac{4}{3}$

Cross-cancel: 6 and 3 share a factor of 3:

$\frac{\cancel{6}^{2}}{1} \times \frac{4}{\cancel{3}_{1}} = \frac{2 \times 4}{1 \times 1} = 8$

This makes sense intuitively: how many $\frac{3}{4}$-sized pieces fit in 6 wholes? Eight of them.

Answer: $8$

Example 5: Divide $\frac{2}{3} \div 4$

$\frac{2}{3} \div \frac{4}{1} = \frac{2}{3} \times \frac{1}{4}$

Cross-cancel: 2 and 4 share a factor of 2:

$\frac{\cancel{2}^{1}}{3} \times \frac{1}{\cancel{4}_{2}} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Dividing Mixed Numbers

To divide mixed numbers, convert them to improper fractions first, then use Keep-Change-Flip.

Example 6: Divide $3\frac{1}{2} \div 1\frac{3}{4}$

Step 1: Convert to improper fractions:

$3\frac{1}{2} = \frac{7}{2} \qquad 1\frac{3}{4} = \frac{7}{4}$

Step 2: Keep-Change-Flip:

$\frac{7}{2} \div \frac{7}{4} = \frac{7}{2} \times \frac{4}{7}$

Step 3: Cross-cancel: the 7s cancel, and 4 and 2 share a factor of 2:

$\frac{\cancel{7}^{1}}{\cancel{2}_{1}} \times \frac{\cancel{4}^{2}}{\cancel{7}_{1}} = \frac{1 \times 2}{1 \times 1} = 2$

Answer: $2$

Example 7: Divide $4\frac{2}{3} \div 1\frac{1}{6}$

Step 1: Convert:

$4\frac{2}{3} = \frac{14}{3} \qquad 1\frac{1}{6} = \frac{7}{6}$

Step 2: Keep-Change-Flip:

$\frac{14}{3} \div \frac{7}{6} = \frac{14}{3} \times \frac{6}{7}$

Step 3: Cross-cancel: 14 and 7 share a factor of 7, and 6 and 3 share a factor of 3:

$\frac{\cancel{14}^{2}}{\cancel{3}_{1}} \times \frac{\cancel{6}^{2}}{\cancel{7}_{1}} = \frac{2 \times 2}{1 \times 1} = 4$

Answer: $4$

Why Does Keep-Change-Flip Work?

Division asks “how many groups of this size fit into that amount?” When you divide by $\frac{3}{4}$, you are asking how many three-quarter-sized pieces fit. Multiplying by the reciprocal $\frac{4}{3}$ answers that question because:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{1}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c}$

Dividing by a number is the same as multiplying by 1 over that number, and 1 over a fraction is just the fraction flipped.

Real-World Application: Carpentry

A carpenter has a plank that is $7\frac{1}{2}$ feet long. She needs to cut it into pieces that are each $1\frac{7}{8}$ feet long. How many pieces can she cut?

Step 1: Convert to improper fractions:

$7\frac{1}{2} = \frac{15}{2} \qquad 1\frac{7}{8} = \frac{15}{8}$

Step 2: Divide:

$\frac{15}{2} \div \frac{15}{8} = \frac{15}{2} \times \frac{8}{15}$

Step 3: Cross-cancel: the 15s cancel, and 8 and 2 share a factor of 2:

$\frac{\cancel{15}^{1}}{\cancel{2}_{1}} \times \frac{\cancel{8}^{4}}{\cancel{15}_{1}} = 4$

The carpenter can cut exactly 4 pieces, each $1\frac{7}{8}$ feet long, from the $7\frac{1}{2}$-foot plank. In practice, she would account for the width of her saw blade (called the “kerf”), which means she might only get 3 full pieces with material left over. Always factor in kerf when planning real cuts.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Keep-Change-Flip: keep the first fraction, change division to multiplication, flip the second fraction
• The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$
• Whole numbers become fractions over 1 before applying the rule
• Mixed numbers must be converted to improper fractions first
• Cross-cancel after flipping to simplify your work
• Division answers “how many of this size fit into that amount?”

Return to Arithmetic for more foundational math topics.