Operations on Fractions

Lessons

Primary 6 · Mathematics

Operations on Fractions

6 lessons

Lesson 5 of 6

Dividing a fraction by a fraction

30 min

Let's begin

Niyonzima roasts 34 kg of groundnuts to sell at school. He packs them into small bags that each hold 18 kg. How many bags can he fill? He needs 34 ÷ 18.

When the pieces are the same size

34 ÷ 14 asks: how many quarters are in 3 quarters? The answer is 3. When the denominators match, just divide the numerators.

For Niyonzima: rename 34 as 68. How many eighths are in 6 eighths? 6 bags. So 34 ÷ 18 = 6.

Why “flip and multiply” works

Dividing by 18 asks how many eighths fit. Every whole holds 8 eighths, so dividing by 18 is the same as multiplying by 8: 34 × 8 = 6. ✔
Dividing by 38 asks how many groups of 3 eighths fit. First find how many eighths (multiply by 8), then group them in threes (divide by 3). That is multiplying by 83, the reciprocal of 38.

The rule: keep the first fraction, change ÷ to ×, and flip the second fraction.

Worked example 1

34 ÷ 38 = 34 × 83 = 2412 = 2. (Two bags of 38 kg fit into 34 kg.)

Worked example 2

56 ÷ 23

Keep, change, flip: 56 × 32.
Multiply: 1512.
Simplify: 1512 = 54 = 1 14.
Check by multiplying back: 1 14 × 23 = 54 × 23 = 1012 = 56. ✔

Worked example 3

(mixed numbers): 2 14 ÷ 1 12

Change to improper fractions: 94 ÷ 32.
Keep, change, flip: 94 × 23.
Cancel (9 with 3 gives 3; 2 with 4 gives 2): 32 × 11 = 32 = 1 12.

Is my answer sensible?

Dividing by a number less than 1 gives an answer bigger than the number you started with, because many small pieces fit. 34 ÷ 18 = 6 is much bigger than 34. Dividing by a number bigger than 1 gives a smaller answer.

Tip

Keep – Change – Flip. Keep the first fraction, change ÷ to ×, flip the second. Then multiply and simplify as usual.

✦

Try it yourself

Work each one in your exercise book first, then tap Show answer to check.

1

45 ÷ 15

Show answerHide answer

Answer: 4. Four fifths contain four fifths.

2

23 ÷ 56

Show answerHide answer

Answer: 23 × 65 = 1215 = 45.

3

78 ÷ 14

Show answerHide answer

Answer: 78 × 4 = 288 = 3 12.

4

3 13 ÷ 2 12

Show answerHide answer

Answer: 103 × 25 = 2015 = 1 13.

5

How many 16 kg packets can be filled from 2 12 kg of sugar?

Show answerHide answer

Answer: 52 × 6 = 302 = 15 packets.

Quick check

5/6 ÷ 1/6 = ?

What you learned today

Same denominators: divide the numerators.
Otherwise keep the first fraction, change ÷ to ×, flip the second fraction.
Dividing by a fraction less than 1 gives a bigger answer; check by multiplying back.

Sources and credits

Grade 6, Unit 4, Lesson 10: Dividing by Unit and Non-Unit Fractions, IM 6–8 Math, authored by Illustrative Mathematics and originally developed by Open Up Resources, licensed under CC BY 4.0. Adapted by Ganzaa.
Grade 6, Unit 4: Dividing Fractions (lessons 1–11), IM 6–8 Math, authored by Illustrative Mathematics and originally developed by Open Up Resources, licensed under CC BY 4.0. Adapted by Ganzaa.
4.3 Multiply and Divide Fractions, from Prealgebra 2e by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax (read on LibreTexts), licensed under CC BY 4.0. Adapted by Ganzaa.

The stories, Rwandan worked examples and practice questions were written by Ganzaa.

1When the pieces are the same size

2Why “flip and multiply” works

3Is my answer sensible?

Try it yourself

When the pieces are the same size

Why “flip and multiply” works

Is my answer sensible?