Dividing a Fraction by a Whole Number

Learn how to divide fractions by whole numbers using visual models and the reciprocal rule!

1. Understanding with Visual Models

The Big Idea

When you divide a fraction by a whole number, you are cutting that fraction into more pieces. When the whole number is greater than 1, the result is smaller than the original fraction!

Example 1: 12÷2\frac{1}{2} \div 2

Think about it: You have half a pie. You share it equally between 2 children. How much does each child get?

Start: 1/21 out of 2 parts shaded÷ 2 → Each part divided into 21 out of 4 parts = 1/4
Remember the Rule:
Dividing by 2 is the same as multiplying by 12\frac{1}{2}
12\frac{1}{2} ÷ 2 = 12\frac{1}{2} × 12\frac{1}{2}
Working:
12\frac{1}{2} ÷ 2
= 12\frac{1}{2} × 12\frac{1}{2}
= 1×12×2\frac{1 × 1}{2 × 2}
= 14\frac{1}{4}
Answer:14\frac{1}{4}

Key Insight: When you cut 12\frac{1}{2} into 2 equal pieces, each piece is 14\frac{1}{4} of the whole.

Example 2: 12÷3\frac{1}{2} \div 3

Context: Mrs Tan bakes a cake. She has half the cake left and shares it equally among 3 children. What fraction of the cake does each child get?

Start: 1/21 out of 2 parts shaded÷ 3 → Each part divided into 31 out of 6 parts = 1/6
Remember the Rule:
Dividing by 3 is the same as multiplying by 13\frac{1}{3}
12\frac{1}{2} ÷ 3 = 12\frac{1}{2} × 13\frac{1}{3}
Working:
12\frac{1}{2} ÷ 3
= 12\frac{1}{2} × 13\frac{1}{3}
= 1×12×3\frac{1 × 1}{2 × 3}
= 16\frac{1}{6}
Answer:16\frac{1}{6}

Example 3: 23÷3\frac{2}{3} \div 3

Start: 2/32 out of 3 parts shaded÷ 3 → Each part divided into 32 out of 9 parts = 2/9
Remember the Rule:
Dividing by 3 is the same as multiplying by 13\frac{1}{3}
23\frac{2}{3} ÷ 3 = 23\frac{2}{3} × 13\frac{1}{3}
Working:
23\frac{2}{3} ÷ 3
= 23\frac{2}{3} × 13\frac{1}{3}
= 2×13×3\frac{2 × 1}{3 × 3}
= 29\frac{2}{9}
Answer:29\frac{2}{9}

2. The Reciprocal Rule

The Magic Rule

Dividing by a number is the same as multiplying by its reciprocal!

34÷2=34×12=38\frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}

What is a Reciprocal?

The reciprocal of a number is 1 divided by that number.

Reciprocal of 2

12\frac{1}{2}

Reciprocal of 3

13\frac{1}{3}

Reciprocal of 4

14\frac{1}{4}

Reciprocal of 5

15\frac{1}{5}

Worked Example: 34÷2\frac{3}{4} \div 2

Start: 3/43 out of 4 parts shaded÷ 2 → Each part divided into 23 out of 8 parts = 3/8
Remember the Rule:
Dividing by 2 is the same as multiplying by 12\frac{1}{2}
34\frac{3}{4} ÷ 2 = 34\frac{3}{4} × 12\frac{1}{2}
Working:
34\frac{3}{4} ÷ 2
= 34\frac{3}{4} × 12\frac{1}{2}
= 3×14×2\frac{3 × 1}{4 × 2}
= 38\frac{3}{8}
Answer:38\frac{3}{8}

3. Practice Problems

Problem 1

13\frac{1}{3} of a cake is shared equally among 4 people. What fraction of the cake does each person get?

ShowHide solution
Start: 1/31 out of 3 parts shaded÷ 4 → Each part divided into 41 out of 12 parts = 1/12
Remember the Rule:
Dividing by 4 is the same as multiplying by 14\frac{1}{4}
13\frac{1}{3} ÷ 4 = 13\frac{1}{3} × 14\frac{1}{4}
Working:
13\frac{1}{3} ÷ 4
= 13\frac{1}{3} × 14\frac{1}{4}
= 1×13×4\frac{1 × 1}{3 × 4}
= 112\frac{1}{12}
Answer:112\frac{1}{12}

Each person gets 112\frac{1}{12} of the cake.

Problem 2

35\frac{3}{5} m of ribbon is cut equally into 6 pieces. What is the length of each piece of ribbon?

ShowHide solution
Start: 3/53 out of 5 parts shaded÷ 6 → Each part divided into 63 out of 30 parts = 1/10
Remember the Rule:
Dividing by 6 is the same as multiplying by 16\frac{1}{6}
35\frac{3}{5} ÷ 6 = 35\frac{3}{5} × 16\frac{1}{6}
Working:
35\frac{3}{5} ÷ 6
= 35\frac{3}{5} × 16\frac{1}{6}
= 3×15×6\frac{3 × 1}{5 × 6}
= 330\frac{3}{30}
= 110\frac{1}{10} (simplified)
Answer:110\frac{1}{10}

Each piece of ribbon is 110\frac{1}{10} m long.

Problem 3: 59÷5\frac{5}{9} \div 5

ShowHide solution
Start: 5/95 out of 9 parts shaded÷ 5 → Each part divided into 55 out of 45 parts = 1/9
Remember the Rule:
Dividing by 5 is the same as multiplying by 15\frac{1}{5}
59\frac{5}{9} ÷ 5 = 59\frac{5}{9} × 15\frac{1}{5}
Working:
59\frac{5}{9} ÷ 5
= 59\frac{5}{9} × 15\frac{1}{5}
= 5×19×5\frac{5 × 1}{9 × 5}
= 545\frac{5}{45}
= 19\frac{1}{9} (simplified)
Answer:19\frac{1}{9}

Problem 4

78\frac{7}{8} kg of sugar is packed equally into 3 packets. What is the mass of each packet of sugar?

ShowHide solution
Start: 7/87 out of 8 parts shaded÷ 3 → Each part divided into 37 out of 24 parts = 7/24
Remember the Rule:
Dividing by 3 is the same as multiplying by 13\frac{1}{3}
78\frac{7}{8} ÷ 3 = 78\frac{7}{8} × 13\frac{1}{3}
Working:
78\frac{7}{8} ÷ 3
= 78\frac{7}{8} × 13\frac{1}{3}
= 7×18×3\frac{7 × 1}{8 × 3}
= 724\frac{7}{24}
Answer:724\frac{7}{24}

Each packet has 724\frac{7}{24} kg of sugar.

Key Takeaways

1.

Dividing a fraction by a whole number greater than 1 makes the result smaller.

2.

The Rule: 34÷2=34×12=38\frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}

3.

Key insight: Dividing by n is the same as multiplying by 1n\frac{1}{n}

4.

Always simplify your answer to the lowest terms!

Watch the video lesson on this topic

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Dividing a Whole Number by a Fraction ›

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