➗ Dividing Fractions

Cambridge Lower Secondary · Grade 7 · Fractions & Decimals

KCF Method
34 ÷ 12  =  34 × 21  = 3/2 = 1½
Keep · Change · Flip
Whole Number ÷ Fraction
6 ÷ 34  =  61 × 43  = 24/3 = 8
8 pieces of length ¾ fit into 6
Mixed Numbers
2½ ÷ 1¼  →  52 ÷ 54  = 2
Convert to improper first!

Watch KCF in action! 🍰

23
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34

What you'll learn:

  • What a reciprocal is and how to find it (flip the fraction)
  • The KCF method: Keep · Change · Flip
  • Dividing a fraction by a fraction
  • Dividing a whole number by a fraction (and vice versa)
  • Dividing mixed numbers (convert first!)
  • Word problems: how many pieces fit?
Created by Mr Josh

📖 Learn: Dividing Fractions

Part 1: Reciprocals

The reciprocal of a fraction is found by flipping it upside down — swapping the numerator and denominator.

📌 Reciprocal of 35  is  53
📌 Reciprocal of 14  is  41 = 4
📌 Reciprocal of 7 → write as 71, flip to get 17
💡 Key fact: a fraction times its reciprocal always equals 1. e.g. 3/5 × 5/3 = 15/15 = 1

Part 2: The KCF Method

To divide fractions, use three steps:

K — Keep   the first fraction exactly as it is
C — Change   the ÷ sign to a × sign
F — Flip   the second fraction (write its reciprocal)
Then multiply as normal: top × top, bottom × bottom. Simplify if needed.
💡 Memory hook: "Keep the first, Change the sign, Flip the last"
ab  ÷  cd  =  ab  ×  dc  =  a×db×c

Part 3: Fraction ÷ Fraction

Example:   23 ÷ 45   →   23 × 54   =   1012   =   5/6
💡 After multiplying, always simplify by dividing numerator and denominator by their HCF.

Part 4: Whole Number ÷ Fraction & Fraction ÷ Whole Number

Whole number ÷ fraction: Write the whole number as a fraction over 1, then KCF.

6 ÷ 34  =  61 × 43  =  243  =  8

Fraction ÷ whole number: Write the whole number as a fraction over 1, then KCF.

56 ÷ 5  =  56 × 15  =  530  =  1/6
💡 Dividing by a whole number n is the same as multiplying by 1/n.

Part 5: Dividing Mixed Numbers

Always convert mixed numbers to improper fractions first, then apply KCF.

Step 1: Convert mixed numbers:   2½ = 52,   1¼ = 54
Step 2: KCF:   52 × 45
Step 3: Multiply:   2010 = 2
💡 To convert: whole × denominator + numerator, keep same denominator. e.g. 2½ → (2×2)+1 = 5, so 5/2
Created by Mr Josh

💡 Worked Examples

Example 1: Fraction ÷ Fraction

Calculate   34 ÷ 25

K — Keep: 34
C — Change ÷ to ×
F — Flip: 25 becomes 52
Multiply: 34 × 52 = 158
Answer: 17/8 (improper → mixed number: 15÷8 = 1 remainder 7)
✅ Dividing by a fraction smaller than 1 gives a bigger result — makes sense!

Example 2: Whole Number ÷ Fraction

Calculate   4 ÷ 23

Step 1: Write 4 as 41
Step 2: KCF: 41 × 32
Step 3: = 122 = 6
💡 "How many two-thirds fit into 4?" → 6. Each whole holds 1½ pieces, and 4 × 1½ = 6. ✓

Example 3: Fraction ÷ Whole Number

Calculate   58 ÷ 5

Step 1: Write 5 as 51
Step 2: KCF: 58 × 15
Step 3: = 540 = 1/8 (divide by HCF 5)
💡 Splitting a fraction into equal groups — each group is smaller.

Example 4: Mixed Number ÷ Mixed Number

Calculate   313 ÷ 123

Step 1: Convert: 3⅓ = 103,   1⅔ = 53
Step 2: KCF: 103 × 35
Step 3: = 3015 = 2
💡 Notice the 3s cancel (cross-cancelling): 10/3 × 3/5 = 10/5 = 2. Spot opportunities to simplify early!
Created by Mr Josh

🎬 Interactive Visualizer

KCF Step Animator

Enter a division problem (fractions as a/b). Watch Keep · Change · Flip unfold!

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23
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34

"How Many Fit?" Number Line

Enter a total length and a piece size (as a fraction). Watch pieces fill the number line one by one!


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Reciprocal Rapid-Fire Quiz ⚡

Type the reciprocal of the fraction shown. 10 questions — go!

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⚠️ Common Mistake Spotlight

When dividing fractions, students sometimes try to divide top by top and bottom by bottom. Here's why that's wrong:

❌ WRONG Method
68 ÷ 34 = ?
Dividing top÷top, bottom÷bottom: 6/3 = 2, 8/4 = 2 → says "answer is 2"

Only works by coincidence here! Fails for most problems.
✅ CORRECT Method (KCF)
68 ÷ 34
= 68 × 43 = 2424 = 1

Always use KCF — it works every time!
Test the wrong method: Try 1/2 ÷ 1/3
Wrong method: 1÷1=1, 2÷3=? (not even a whole number!)
KCF: ½ × 3/1 = 3/2 = 1½ ✅
Created by Mr Josh

✏️ Exercise 1: Finding Reciprocals

Type the reciprocal of each fraction in the form a/b. If the answer is a whole number, just type the number.

Created by Mr Josh

✏️ Exercise 2: Fraction ÷ Fraction

Use KCF to divide. Give your answer as a fraction in its simplest form (e.g. 3/4) or a whole/mixed number.

Created by Mr Josh

✏️ Exercise 3: Whole ÷ Fraction & Fraction ÷ Whole

Remember to write whole numbers as fractions over 1 first, then KCF.

Created by Mr Josh

✏️ Exercise 4: Dividing Mixed Numbers

Convert to improper fractions first, then apply KCF. Give answers as mixed numbers or simplified fractions.

Created by Mr Josh

📏 Exercise 5: Word Problems

Set up a division, apply KCF, and answer the question. Give answers as whole numbers, mixed numbers, or simplified fractions.

Created by Mr Josh

📝 Practice Questions

  1. Write the reciprocal of 4/7.
  2. Calculate: 1/2 ÷ 1/4
  3. Calculate: 3/5 ÷ 3/10
  4. Calculate: 7/8 ÷ 1/4
  5. Calculate: 2/3 ÷ 4/9
  6. Calculate: 5 ÷ 1/3
  7. Calculate: 8 ÷ 2/5
  8. Calculate: 3/4 ÷ 6
  9. Calculate: 5/6 ÷ 10
  10. Calculate: 12 ÷ 3/4
  11. Calculate: 11/2 ÷ 3/4
  12. Calculate: 22/3 ÷ 4/3
  13. Calculate: 31/4 ÷ 15/8
  14. How many pieces of length 1/3 m can be cut from a 4 m rope?
  15. A recipe needs 3/4 cup of flour per batch. You have 6 cups. How many batches can you make?
  16. A car travels 2/5 km per minute. How many minutes to travel 6 km?
  17. What is the reciprocal of 9?
  18. Simplify: 6/8 ÷ 3/4
  19. Calculate: 21/2 ÷ 11/4
  20. A path is 71/2 m long. Fence posts are placed every 3/4 m. How many gaps are there?
  1. 7/4
  2. ½ × 4/1 = 4/2 = 2
  3. 3/5 × 10/3 = 30/15 = 2
  4. 7/8 × 4/1 = 28/8 = 3½
  5. 2/3 × 9/4 = 18/12 = 3/2 = 1½
  6. 5/1 × 3/1 = 15
  7. 8/1 × 5/2 = 40/2 = 20
  8. 3/4 × 1/6 = 3/24 = 1/8
  9. 5/6 × 1/10 = 5/60 = 1/12
  10. 12/1 × 4/3 = 48/3 = 16
  11. 3/2 × 4/3 = 12/6 = 2
  12. 8/3 × 3/4 = 24/12 = 2
  13. 13/4 × 8/13 = 104/52 = 2
  14. 4 ÷ 1/3 = 4 × 3 = 12 pieces
  15. 6 ÷ 3/4 = 6 × 4/3 = 24/3 = 8 batches
  16. 6 ÷ 2/5 = 6 × 5/2 = 30/2 = 15 minutes
  17. 1/9
  18. 6/8 × 4/3 = 24/24 = 1
  19. 5/2 × 4/5 = 20/10 = 2
  20. 15/2 × 4/3 = 60/6 = 10 gaps
Created by Mr Josh

🏆 Challenge: Multi-Step & Tricky Problems

  1. A ribbon is 41/2 m long. It is cut into pieces of 3/8 m each. How many full pieces can be cut?
  2. Evaluate: (2/3 ÷ 4/9) × 3/4. Give your answer in simplest form.
  3. A tank fills at a rate of 5/6 of its volume per hour. How many hours to fill 21/2 tanks?
  4. Three equal pieces are cut from a plank of length 41/2 m. What is the length of each piece? (Use division.)
  5. Amira divides 5/6 of her savings equally among 5 friends and 2 charity boxes (7 equal parts). What fraction of her savings does each part receive?
  6. Which is larger: (3 ÷ 3/5) or (21/2 ÷ 1/4)? By how much?
  7. A sequence of fractions: 1/2, 1/4, 1/8, … Each term is divided by 2 (i.e. multiplied by 1/2). What is the 6th term?
  8. A car uses 3/8 of a tank of fuel to travel 21/4 km. How far can it travel on a full tank?
  1. 9/2 ÷ 3/8 = 9/2 × 8/3 = 72/6 = 12 pieces
  2. 2/3 ÷ 4/9 = 2/3 × 9/4 = 18/12 = 3/2. Then 3/2 × 3/4 = 9/8 = 1⅛
  3. 5/2 ÷ 5/6 = 5/2 × 6/5 = 30/10 = 3 hours
  4. 9/2 ÷ 3 = 9/2 × 1/3 = 9/6 = 1½ m
  5. 5/6 ÷ 7 = 5/6 × 1/7 = 5/42 of her savings
  6. 3 ÷ 3/5 = 3 × 5/3 = 5;   5/2 ÷ 1/4 = 5/2 × 4 = 10. So 10 is larger, by 10 − 5 = 5
  7. ½ → ¼ → ⅛ → 1/16 → 1/32 → 1/64
  8. Full tank distance = 9/4 ÷ 3/8 = 9/4 × 8/3 = 72/12 = 6 km
Created by Mr Josh