Lesson 9 ~30 min Unit 1 · Fractions +90 XP

Multiplying and Dividing Fractions

Multiply straight across the top and bottom. Divide? Flip the second fraction and multiply. It is that simple.

Today’s hook: Half of a half is a quarter. So $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. Multiplying fractions makes things smaller. And dividing? $\frac{1}{2} \div \frac{1}{4}$ means how many quarters fit in a half. The answer is 2!

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Think First

warm-up

Before you read on — quickly: What is half of a third? And how many halves fit in three wholes? Write your guesses.

Record your answer in your workbook.

The Big Idea

+5 XP

Multiplying fractions: multiply the numerators, multiply the denominators. Dividing fractions: flip the second fraction (find its reciprocal) and multiply. Mixed numbers: convert to improper fractions first, then multiply or divide.

Multiplying fractions is easier than adding them. No common denominator needed. Just multiply straight across: $\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15} = \frac{2}{5}$. For division: keep, change, flip. Keep the first fraction, change ÷ to ×, flip the second. $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$.

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ and $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

No common denominator

Unlike addition, multiplication needs no LCD. Just multiply across.

Keep, change, flip

Keep first, change ÷ to ×, flip the second fraction.

Simplify early

Cross-cancel before multiplying to keep numbers small.

What You’ll Master

objectives

Know

Multiply numerators and denominators straight across
Division = multiply by the reciprocal
‘Of’ means multiply

Understand

Why multiplying fractions makes numbers smaller
Why dividing by a fraction gives a larger answer
Why the reciprocal works for division

Can Do

Multiply any two fractions and simplify
Divide any two fractions using the reciprocal
Handle mixed numbers in multiplication and division

Words You Need

vocabulary

Multiply acrossMultiply the top numbers together and the bottom numbers together.

ReciprocalFlip a fraction upside down. The reciprocal of a/b is b/a.

Keep, change, flipThe three steps for dividing fractions: keep first, change to ×, flip second.

Of means ×In maths, ‘half of 6’ means 1/2 × 6. ‘Of’ always means multiply.

Cross-cancelSimplify diagonally before multiplying to make calculation easier.

Convert firstAlways change mixed numbers to improper fractions before multiplying or dividing.

Spot the Trap

heads-up

Wrong: $\frac{2}{3} \times \frac{1}{4} = \frac{2}{3} + \frac{1}{4}$ first. No! Multiplication is a completely different operation. Multiply straight across.

Right: $\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}$. Multiply tops together, bottoms together.

Wrong: $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$. No! You flipped the wrong fraction. Flip the second fraction.

Right: $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$. Keep, change, flip the second fraction only.

Multiplying Fractions

+5 XP

Multiply the numerators together. Multiply the denominators together. Simplify the result. Cross-cancel before multiplying to make numbers smaller and easier.

Calculate $\frac{3}{4} \times \frac{8}{9}$. Cross-cancel first: 3 and 9 share factor 3, 4 and 8 share factor 4. $\frac{3 \div 3}{4 \div 4} \times \frac{8 \div 4}{9 \div 3} = \frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$. Much easier than $\frac{24}{36}$ then simplifying!

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Cross-cancel first

Makes numbers smaller and eliminates simplifying later.

No LCD needed

Unlike addition, just multiply straight across. No common denominator.

Whole numbers too

$\frac{2}{3} \times 5 = \frac{2}{3} \times \frac{5}{1} = \frac{10}{3}$. Write whole numbers over 1.

Dividing Fractions

+5 XP

Keep, change, flip. Keep the first fraction exactly as it is. Change the division sign to multiplication. Flip the second fraction upside down (find its reciprocal). Then multiply as normal.

Calculate $\frac{3}{5} \div \frac{9}{10}$. Keep 3/5. Change ÷ to ×. Flip 9/10 to 10/9. Now multiply: $\frac{3}{5} \times \frac{10}{9}$. Cross-cancel: 3 and 9 share 3, 5 and 10 share 5. $\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Flip the second only

Never flip the first fraction. Only the one after the division sign.

Dividing by a fraction

Makes the answer bigger. How many pieces fit? More than you started with.

Then cross-cancel

After flipping, cross-cancel before the final multiplication step.

Mixed Numbers

+5 XP

Always convert mixed numbers to improper fractions first. Then multiply or divide as normal. Convert your final answer back to a mixed number if needed.

Calculate $2\frac{1}{2} \times 1\frac{1}{3}$. Convert: $2\frac{1}{2} = \frac{5}{2}$ and $1\frac{1}{3} = \frac{4}{3}$. Multiply: $\frac{5}{2} \times \frac{4}{3} = \frac{20}{6}$. Simplify: $\frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$.

$a\frac{b}{c} \times d\frac{e}{f} = \frac{ca+b}{c} \times \frac{fd+e}{f}$

Convert first

Never multiply mixed numbers directly. Improper fractions only.

Convert back at the end

If the answer is an improper fraction, write it as a mixed number.

Same for division

Convert both mixed numbers, then keep-change-flip and multiply.

Watch Me Solve It · 3 examples

step-by-step

Example 1: Multiplying two fractions

Calculate $\frac{4}{7} \times \frac{21}{16}$

Cross-cancel first. 4 and 16 share factor 4: 4÷4=1, 16÷4=4. 7 and 21 share factor 7: 7÷7=1, 21÷7=3.

Multiply the simplified fractions: $\frac{1}{1} \times \frac{3}{4} = \frac{1 \times 3}{1 \times 4} = \frac{3}{4}$.

Check: HCF(3, 4) = 1, so fully simplified. Without cancelling first: $\frac{84}{112}$, then HCF = 28, giving the same answer.

$\frac{4}{7} \times \frac{21}{16} = \frac{3}{4}$

Example 2: Dividing fractions

Calculate $\frac{5}{8} \div \frac{15}{16}$

Keep, change, flip. Keep $\frac{5}{8}$. Change ÷ to ×. Flip $\frac{15}{16}$ to $\frac{16}{15}$. So: $\frac{5}{8} \times \frac{16}{15}$.

Cross-cancel: 5 and 15 share 5, 8 and 16 share 8. $\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$.

Check: Does $\frac{2}{3} \times \frac{15}{16} = \frac{5}{8}$? $\frac{30}{48} = \frac{5}{8}$. Yes!

$\frac{5}{8} \div \frac{15}{16} = \frac{2}{3}$

Example 3: Mixed number division

Calculate $3\frac{1}{3} \div 1\frac{2}{3}$

Convert to improper fractions. $3\frac{1}{3} = \frac{10}{3}$ and $1\frac{2}{3} = \frac{5}{3}$.

Keep, change, flip: $\frac{10}{3} \times \frac{3}{5}$. Cross-cancel: 3 and 3 share 3, 10 and 5 share 5. $\frac{2}{1} \times \frac{1}{1} = \frac{2}{1}$.

Result is $\frac{2}{1} = 2$. Since the answer is a whole number, no mixed number needed.

$3\frac{1}{3} \div 1\frac{2}{3} = 2$

Common Pitfalls

avoid these

Mistake: Finding a common denominator for multiplication. No! Only addition/subtraction need LCD. Multiply straight across.

Fix: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$. No LCD needed. Just multiply tops and bottoms.

Mistake: Flipping the first fraction in division. $\frac{1}{2} \div \frac{1}{3} \ne \frac{2}{1} \times \frac{1}{3}$. You flipped the wrong one!

Fix: Keep the first, flip the second. $\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} = 1\frac{1}{2}$.

Mistake: Multiplying mixed numbers without converting. $2\frac{1}{2} \times 1\frac{1}{2} \ne 2\frac{1}{4}$. Whole numbers don’t multiply separately!

Fix: Convert first: $\frac{5}{2} \times \frac{3}{2} = \frac{15}{4} = 3\frac{3}{4}$.

Copy Into Your Books

essential notes

Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ (straight across, no LCD needed)

Divide: keep, change, flip. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Cross-cancel before multiplying to keep numbers small.

Mixed numbers: convert to improper first, then multiply/divide.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Multiply & Divide

4 problems

Four drill problems to build your fraction multiplication and division fluency. Work each, then reveal the answer.

1 $\frac{2}{5} \times \frac{15}{8}$

Cross-cancel: 2 and 8 share 2, 5 and 15 share 5. 1/1 × 3/4 = 3/4.3/4
2 $\frac{7}{10} \div \frac{14}{15}$

Keep, change, flip: 7/10 × 15/14. Cross-cancel: 7 and 14 share 7, 10 and 15 share 5. 1/2 × 3/1 = 3/4.3/4
3 $\frac{3}{4} \times 12$

Write 12 as 12/1. 3/4 × 12/1 = 36/4 = 9. Or: 3/4 of 12 = 9.9
4 $2\frac{1}{4} \div 1\frac{1}{2}$

Convert: 9/4 ÷ 3/2. Keep, change, flip: 9/4 × 2/3. Cross-cancel: 9 and 3 share 3, 4 and 2 share 2. 3/2 × 1/1 = 3/2 = 1 1/2.1 1/2

Complete in your workbook.

Quick Check · 5 questions

$\frac{2}{3} \times \frac{9}{10} = ?$

+10 XP

$\frac{3}{4} \div \frac{3}{8} = ?$

+10 XP

$\frac{2}{3} \times 6 = ?$

+10 XP

$1\frac{3}{4} \div 1\frac{1}{2} = ?$

+10 XP

$\frac{5}{6} \div \frac{5}{12} = ?$

+10 XP

Show Your Working · 3 questions

Show Your Working

11 marks total

ApplyMedium4 MARKS

Q6. (a) Calculate $\frac{8}{15} \times \frac{25}{32}$. Use cross-cancelling and show each step. (b) Calculate $\frac{9}{14} \div \frac{3}{7}$. Show keep-change-flip.

Answer in your workbook.

ApplyMedium4 MARKS

Q7. A rectangle has length $2\frac{1}{2}$ m and width $\frac{3}{5}$ m. (a) Find the area. (b) If the rectangle is divided into equal strips of width $\frac{1}{4}$ m, how many strips fit along the length?

Answer in your workbook.

ReasonHard3 MARKS

Q8. A student says $\frac{1}{2} \div \frac{1}{4} = \frac{1}{8}$ because you just divide the denominators. Explain why this is wrong, and show the correct working.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — 2/3 × 9/10 = 18/30 = 3/5 (HCF = 6).

2. C — 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2.

3. B — 2/3 × 6/1 = 12/3 = 4.

4. D — 7/4 ÷ 3/2 = 7/4 × 2/3 = 14/12 = 7/6 = 1 1/6.

5. A — 5/6 ÷ 5/12 = 5/6 × 12/5 = 60/30 = 2.

Show Your Working Model Answers

Q6 (4 marks): (a) Cross-cancel: 8 and 32 share 8, 15 and 25 share 5 [1]. 1/3 × 5/4 = 5/12 [1]. (b) Keep 9/14, change to ×, flip to 7/3 [1]. 9/14 × 7/3 = 63/42 = 3/2 = 1 1/2 [1].

Q7 (4 marks): (a) 2 1/2 = 5/2 [0.5]. Area = 5/2 × 3/5 = 15/10 = 1 1/2 m² [1.5]. (b) 5/2 ÷ 1/4 = 5/2 × 4/1 = 20/2 = 10 strips [2].

Q8 (3 marks): Cannot just divide denominators [1]. Correct: keep-change-flip: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2 [1]. Dividing by a smaller fraction gives a bigger answer [1].

Stretch Challenge · +25 XP, +10 coins

The Fraction Maze

Start with $\frac{16}{27}$. Apply each operation in order: (1) multiply by $\frac{9}{8}$, (2) divide by $\frac{2}{3}$, (3) multiply by $\frac{3}{4}$. What is your final answer? Can you find a single fraction that would get you from start to finish in one step?

Reveal solution

Step 1: $\frac{16}{27} \times \frac{9}{8} = \frac{2}{3}$ (cancel 16÷8, 27÷9). Step 2: $\frac{2}{3} \div \frac{2}{3} = \frac{2}{3} \times \frac{3}{2} = 1$. Step 3: $1 \times \frac{3}{4} = \frac{3}{4}$. One-step: $\frac{16}{27} \times \frac{9}{8} \div \frac{2}{3} \times \frac{3}{4} = \frac{16}{27} \times \frac{9}{8} \times \frac{3}{2} \times \frac{3}{4} = \frac{3}{4}$. The single fraction is $\frac{81}{144} = \frac{3}{4}$, but more elegantly: multiply by $\frac{81}{144}$ or simply $\frac{3}{4}$ total effect.

Quick Review

Multiply

Top × top, bottom × bottom

Divide

Keep, change, flip

Cross-cancel

Before multiplying

No LCD

Unlike addition

Mixed numbers

Convert to improper first

Of means ×

Half of = 1/2 ×

Interactive: Fraction Operations

Explore fraction multiplication and division with interactive visual models. See how keep-change-flip works step by step.

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