Chapter 3.19: Multiplying Fractions by Fractions

🎯 Learning Journey

Step 1: Multiply the Numerators

Multiply the top numbers (numerators) together to get the numerator of your answer.

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Step 2: Multiply the Denominators

Multiply the bottom numbers (denominators) together to get the denominator of your answer.

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Step 3: Simplify Your Answer

Find the highest common factor (HCF) of the numerator and denominator, then divide both by it to simplify to lowest terms.

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Step 4: Check Your Work

Verify that your answer is in simplest form and makes sense in context. Remember: multiplying fractions makes the answer smaller!

📖 Understanding the Topic

🎯 What You'll Learn

When multiplying fractions by fractions, you simply multiply the numerators together and the denominators together. The key rule to remember is: multiply straight across! For example, ½ × ¾ means "½ of ¾", which equals 3/8. Unlike adding or subtracting fractions, you don't need common denominators.

🚀 Why This Matters

Understanding "Of"

Finding a fraction "of" something means multiplication. This appears constantly in real-world problems.

Scaling and Proportion

Essential for recipes, scaling drawings, calculating discounts, and working with proportions in science.

Algebraic Foundation

Multiplying fractions is crucial for solving algebraic equations and working with rational expressions.

💡 Worked Examples

Example 1: Baking Recipe

A recipe calls for ½ cup of sugar, but you only want to make ¾ of the recipe. How much sugar do you need?

½ × ¾ = ?

Multiply numerators: 1 × 3 = 3

Multiply denominators: 2 × 4 = 8

Answer: 3/8 cup of sugar

Example 2: Paint Mixing

You need 2/3 of a tin of paint, but you only want to use 3/4 of that amount. What fraction do you actually use?

2/3 × 3/4 = ?

Multiply numerators: 2 × 3 = 6

Multiply denominators: 3 × 4 = 12

Simplify: 6/12 = 1/2 (÷ 6)

Answer: 1/2 tin of paint

Example 3: Garden Plot

A garden is 3/5 of an acre. You plant vegetables in 2/3 of this area. What fraction of an acre is vegetables?

3/5 × 2/3 = ?

Multiply numerators: 3 × 2 = 6

Multiply denominators: 5 × 3 = 15

Simplify: 6/15 = 2/5 (÷ 3)

Answer: 2/5 of an acre

Example 4: Time Management

You spend 3/4 of an hour studying. Of that time, 4/5 is spent on maths. What fraction of an hour is maths?

3/4 × 4/5 = ?

Multiply numerators: 3 × 4 = 12

Multiply denominators: 4 × 5 = 20

Simplify: 12/20 = 3/5 (÷ 4)

Answer: 3/5 of an hour

✏️ Practice Questions

Question 1: Calculate 1/2 × 1/3

1/5

1/6

2/5

2/6

Question 2: What is 2/3 × 3/4 in simplest form?

6/12

1/2

5/7

2/3

Question 3: Calculate 3/5 × 2/7

5/12

6/35

1/2

6/12

Question 4: What is 4/5 × 5/8 in simplest form?

20/40

1/2

9/13

2/3

Question 5: A cake recipe needs 3/4 cup flour. You make 2/3 of the recipe. How much flour?

5/7 cup

1/2 cup

6/12 cup

5/12 cup

Question 6: Calculate 5/6 × 3/10 in simplest form

15/60

1/4

8/16

1/3

Question 7: What is 2/5 × 5/6 in simplest form?

10/30

1/3

7/11

2/6

Question 8: Calculate 7/8 × 4/7 in simplest form

28/56

1/2

11/15

4/8

⚠️ Common Mistakes & How to Avoid Them

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

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Adding Instead of Multiplying

What students often do wrong:

Students confuse multiplication with addition and add numerators and denominators separately (1/2 × 1/3 = 2/5 ✗), or they try to find common denominators as if adding fractions.

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How to Avoid This Mistake

Correct approach: Multiply straight across - numerator × numerator, denominator × denominator. No common denominators needed!

Memory tip: "Multiply fractions? Straight across!" - always multiply tops together and bottoms together.

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Forgetting to Simplify

What students often do wrong:

After multiplying, students leave answers like 6/12 or 15/20 without simplifying to lowest terms (1/2 and 3/4 respectively).

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How to Avoid This Mistake

Correct approach: Always check if numerator and denominator share common factors. Divide both by the HCF to simplify completely.

Memory tip: "Simplify makes it pretty" - always finish by simplifying to lowest terms.

💡 Teacher's Tip

You can also simplify before multiplying! Look for common factors diagonally - if a numerator and denominator share a factor, cancel them first. For example: 4/5 × 5/8 - cancel the 5s to get 4/8, then simplify to 1/2. This makes calculations easier!

📋 Chapter Summary

🎉 Congratulations!

You've mastered Multiplying Fractions by Fractions!

🎯 Skills You've Developed:

✓ Multiply numerators and denominators correctly

✓ Simplify fraction products to lowest terms

✓ Apply cancellation before multiplying when possible

✓ Solve real-world problems involving fraction multiplication

✓ Understand that multiplying fractions makes answers smaller

🚀 What's Next?

Next: Learn to divide fractions by fractions using the "keep, change, flip" method