Chapter 3.20: Dividing Fractions by Fractions

🎯 Learning Journey

Step 1: KEEP the First Fraction

Write down the first fraction exactly as it is. This fraction stays the same throughout the process.

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Step 2: CHANGE Division to Multiplication

Replace the division sign (÷) with a multiplication sign (×). This is the crucial step in the method.

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Step 3: FLIP the Second Fraction

Find the reciprocal of the second fraction by flipping it upside down (swap numerator and denominator).

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Step 4: Multiply and Simplify

Now multiply the fractions (numerator × numerator, denominator × denominator) and simplify your answer to lowest terms.

📖 Understanding the Topic

🎯 What You'll Learn

Dividing fractions uses the "Keep, Change, Flip" (KCF) method. When you divide by a fraction, you're actually asking "how many of the second fraction fit into the first?" The trick is to multiply by the reciprocal instead. For example, ½ ÷ ¼ asks "how many quarters are in a half?" The answer is 2, which we get by calculating ½ × 4/1 = 2/1 = 2.

🚀 Why This Matters

Sharing and Distribution

Essential for solving problems about dividing quantities into fractional parts, like sharing resources or portioning ingredients.

Rate and Ratio Problems

Used constantly in calculating speeds, prices per unit, and efficiency in real-world contexts.

Advanced Mathematics

Critical foundation for algebra, solving equations with fractions, and understanding rational functions.

💡 Worked Examples

Example 1: Ribbon Cutting

You have ¾ metre of ribbon. How many ⅛ metre pieces can you cut from it?

¾ ÷ ⅛ = ?

KEEP: ¾

CHANGE: ÷ becomes ×

FLIP: ⅛ becomes 8/1

Multiply: ¾ × 8/1 = 24/4 = 6

Answer: 6 pieces

Example 2: Pizza Sharing

You have ⅖ of a pizza. If you divide it into servings of ⅕, how many servings do you have?

⅖ ÷ ⅕ = ?

KEEP: ⅖

CHANGE: ÷ becomes ×

FLIP: ⅕ becomes 5/1

Multiply: ⅖ × 5/1 = 10/5 = 2

Answer: 2 servings

Example 3: Paint Portions

You need to divide ⅔ litre of paint into containers holding ⅙ litre each. How many containers?

⅔ ÷ ⅙ = ?

KEEP: ⅔

CHANGE: ÷ becomes ×

FLIP: ⅙ becomes 6/1

Multiply: ⅔ × 6/1 = 12/3 = 4

Answer: 4 containers

Example 4: Fabric Division

A tailor has ⅘ metre of fabric. Each project needs ⅖ metre. How many projects can be made?

⅘ ÷ ⅖ = ?

KEEP: ⅘

CHANGE: ÷ becomes ×

FLIP: ⅖ becomes 5/2

Multiply: ⅘ × 5/2 = 20/10 = 2

Answer: 2 projects

✏️ Practice Questions

Question 1: Calculate ½ ÷ ¼

1

2

⅛

4

Question 2: What is ⅔ ÷ ⅙ in simplest form?

1

4

4/18

3

Question 3: Calculate ¾ ÷ ⅜

1

2

3/32

½

Question 4: What is ⅘ ÷ ⅖ in simplest form?

⅖

2

8/10

1

Question 5: A recipe needs ⅔ cup sugar. If you divide it into ⅙ cup portions, how many portions?

2

4

6

3

Question 6: Calculate 5/6 ÷ ⅓ in simplest form

5/18

5/2 or 2½

2

3

Question 7: What is ⅖ ÷ ⅕?

1

2

4/25

5

Question 8: Calculate 7/8 ÷ ¼ in simplest form

7/32

7/2 or 3½

3

4

⚠️ Common Mistakes & How to Avoid Them

Learn from typical errors students make and discover how to avoid them!

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Flipping the Wrong Fraction

What students often do wrong:

Students flip the first fraction instead of the second, or flip both fractions, leading to incorrect answers. For ½ ÷ ¼, they might calculate 2/1 ÷ 4/1 = ½ ✗

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How to Avoid This Mistake

Correct approach: Remember "Keep, Change, Flip" - the first fraction KEEPS its place. Only the SECOND fraction gets flipped!

Memory tip: "First stays, second flips the other way" - only flip the dividing fraction.

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Forgetting to Change the Operation

What students often do wrong:

Students flip the second fraction but forget to change the division sign to multiplication, attempting to divide by the reciprocal instead of multiplying.

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How to Avoid This Mistake

Correct approach: Always do all three steps: Keep the first fraction, Change ÷ to ×, Flip the second fraction. Then multiply!

Memory tip: "KCF then multiply" - complete all steps before calculating.

💡 Teacher's Tip

Check your answer makes sense! When dividing by a fraction less than 1, your answer should be LARGER than the first fraction. For example, ½ ÷ ¼ = 2, which is larger than ½. This is because you're asking "how many quarters fit into a half?"

📋 Chapter Summary

🎉 Congratulations!

You've mastered Dividing Fractions by Fractions!

🎯 Skills You've Developed:

✓ Apply the "Keep, Change, Flip" method correctly

✓ Find reciprocals of fractions accurately

✓ Convert division problems to multiplication

✓ Simplify answers to lowest terms

✓ Solve real-world problems involving fraction division

🚀 What's Next?

Next: Explore statistics - learn to construct and interpret data displays