Lesson 7 ~25 min Unit 1 · Fractions +85 XP

Equivalent Fractions and Simplifying

Multiply or divide top and bottom by the same number. Find the simplest form. Master the cancelling game.

Today's hook: Is $\frac{1}{2}$ the same as $\frac{2}{4}$? Of course — same amount of pizza, just cut differently. The magic rule: whatever you do to the top, you must do to the bottom.

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Think First

warm-up

Before you read on — quickly: Is $\frac{2}{3}$ the same as $\frac{4}{6}$? How can you prove it without a calculator?

Record your answer in your workbook.

The Big Idea

+5 XP

Equivalent fractions represent the same value. You can create them by multiplying or dividing the numerator AND denominator by the same number. Simplifying means finding the simplest form using the highest common factor.

Think of equivalent fractions as the same pizza cut into different numbers of slices. $\frac{1}{2}$ = $\frac{2}{4}$ = $\frac{3}{6}$ = $\frac{4}{8}$ — all the same amount. The rule: whatever you do to the top, do to the bottom. To simplify: divide both by their HCF (highest common factor).

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

Same pizza, different cuts

Equivalent fractions = same amount, different names.

Whatever you do to the top

Do the same to the bottom. Always. No exceptions.

Infinitely many equivalents

Multiply by 2, 3, 4, 5... forever. All are valid.

What You'll Master

objectives

Know

Equivalent fractions have the same value
HCF is used for simplifying
Cross-multiplication checks equivalence

Understand

Why multiplying/dividing both parts equally works
Why simplest form has no common factors
How cancelling works before multiplication

Can Do

Create equivalent fractions and simplify to lowest terms
Use cross-multiplication to check equivalence
Cancel diagonally before multiplying

Words You Need

vocabulary

Equivalent fractionFractions that have the same value but different numerators and denominators.

SimplifyReduce a fraction to its simplest form by dividing by the HCF.

Lowest termsWhen numerator and denominator have no common factors other than 1.

HCFHighest Common Factor. The largest number that divides both numbers.

CancellingDividing numerator and denominator by a common factor before multiplying.

Cross-cancelCancelling diagonally across a multiplication sign before multiplying.

Spot the Trap

heads-up

Wrong: "Add to both parts to get an equivalent." No! You multiply or divide, not add. $\frac{1}{2} \ne \frac{1+1}{2+1} = \frac{2}{3}$.

Right: Whatever you do to the top, do the same to the bottom. Multiply or divide only. $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Wrong: $\frac{4}{6}$ simplified is $\frac{2}{3}$ by subtracting 2. No! You divide both by 2, not subtract.

Right: Divide both numerator AND denominator by the HCF. $\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$.

Equivalent Fractions

+5 XP

Equivalent fractions have the same value. Multiply or divide numerator and denominator by the same number. Cross-multiply to check: $\frac{a}{b} = \frac{c}{d}$ if and only if $a \times d = b \times c$.

Are $\frac{2}{3}$ and $\frac{8}{12}$ equivalent? Cross-multiply: 2 × 12 = 24, 3 × 8 = 24. Since both equal 24, they are equivalent. You can also see: $\frac{2}{3} \times \frac{4}{4} = \frac{8}{12}$. Multiplying by $\frac{4}{4}$ = 1 doesn't change the value.

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

Think of it as pizza

Same amount, just cut into more or fewer slices.

Cross-multiply to check

If a×d = b×c, they're equivalent. Quick verification.

× 1 doesn't change value

$\frac{2}{2}$, $\frac{3}{3}$, $\frac{4}{4}$ all equal 1. Multiplying by 1 is safe.

Simplifying

+5 XP

Simplifying means finding the simplest form by dividing numerator and denominator by their HCF. If you can't find the HCF, divide by any common factor repeatedly.

Simplify $\frac{48}{72}$. HCF of 48 and 72 = 24. Divide both by 24: $\frac{48 \div 24}{72 \div 24} = \frac{2}{3}$. Check: 2 and 3 have no common factors (HCF = 1), so $\frac{2}{3}$ is fully simplified. If both are even, start by dividing by 2.

$\frac{48}{72} = \frac{48 \div 24}{72 \div 24} = \frac{2}{3}$

Find the HCF

Divide by the largest common factor for one-step simplification.

Or divide repeatedly

If HCF is hard to find, divide by 2, then 3, then 5...

Simplest form check

If both are even, you can divide by 2 again. Keep going!

The Cancelling Shortcut

+5 XP

Before multiplying fractions, you can cross-cancel diagonally. This keeps numbers smaller and makes multiplication easier. Only cancel across a multiplication sign.

Before calculating $\frac{2}{3} \times \frac{9}{10}$, look diagonally: 2 and 10 share factor 2, 3 and 9 share factor 3. Cancel: $\frac{2}{3} \times \frac{9}{10} = \frac{1}{1} \times \frac{3}{5} = \frac{3}{5}$. Much easier than $\frac{18}{30}$ then simplifying!

$\frac{2}{3} \times \frac{9}{10} = \frac{1}{1} \times \frac{3}{5} = \frac{3}{5}$

Look diagonally

Top-left with bottom-right, bottom-left with top-right.

Cancel before multiplying

Smaller numbers = easier calculation and no simplifying needed.

Only across ×

Never cancel across + or −. Only works for multiplication.

Watch Me Solve It · 3 examples

step-by-step

Example 1: Finding an equivalent fraction

Find the missing number: $\frac{3}{5} = \frac{?}{20}$

What did 5 become? 5 × 4 = 20. So we multiplied the bottom by 4.

Do the same to the top: 3 × 4 = 12.

Check: $\frac{3}{5} \times \frac{4}{4} = \frac{12}{20}$. Since $\frac{4}{4} = 1$, the value is unchanged.

$\frac{3}{5} = \frac{12}{20}$

Example 2: Simplifying to lowest terms

Simplify $\frac{36}{48}$ fully.

Find the HCF of 36 and 48. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Common: 1, 2, 3, 4, 6, 12. HCF = 12.

Divide both by 12: $\frac{36 \div 12}{48 \div 12} = \frac{3}{4}$.

Check: HCF(3, 4) = 1, so no more simplifying possible.

$\frac{36}{48} = \frac{3}{4}$

Example 3: Cross-cancelling before multiplication

Simplify $\frac{4}{9} \times \frac{3}{8}$ using cross-cancellation.

Look diagonally: 4 and 8 share factor 4. 9 and 3 share factor 3.

Cancel: $\frac{4 \div 4}{9 \div 3} \times \frac{3 \div 3}{8 \div 4} = \frac{1}{3} \times \frac{1}{2}$.

Multiply the simplified fractions: $\frac{1 \times 1}{3 \times 2} = \frac{1}{6}$.

$\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}$

Common Pitfalls

avoid these

Mistake: Adding the same number to numerator and denominator. $\frac{1}{2} \ne \frac{1+3}{2+3} = \frac{4}{5}$. This changes the value!

Fix: Only multiply or divide both parts by the same number. $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.

Mistake: Simplifying only the numerator or only the denominator. $\frac{6 \div 2}{8} = \frac{3}{8}$ is wrong — you must divide both!

Fix: Divide both top and bottom by the same factor. $\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$.

Mistake: Cancelling across addition. $\frac{2}{3} + \frac{1}{3} \ne \frac{2}{1} + \frac{1}{1}$. Cancelling only works for multiplication!

Fix: Only cross-cancel across a × sign. For addition, find a common denominator first.

Copy Into Your Books

essential notes

Equivalent fractions: $\frac{a}{b} = \frac{a \times c}{b \times c} = \frac{a \div c}{b \div c}$

Simplify by dividing both parts by their HCF (highest common factor).

Cross-multiplication test: $\frac{a}{b} = \frac{c}{d}$ if $a \times d = b \times c$.

Cross-cancel: only across ×, look diagonally, divide by common factors before multiplying.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Equivalent & Simplify

4 problems

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

1 Find the missing number: $\frac{2}{7} = \frac{?}{28}$

7 × 4 = 28, so multiply top by 4: 2 × 4 = 8.8
2 Simplify $\frac{56}{72}$ fully.

HCF(56, 72) = 8. 56 ÷ 8 = 7, 72 ÷ 8 = 9. Check: HCF(7, 9) = 1.7/9
3 Are $\frac{5}{8}$ and $\frac{15}{24}$ equivalent? Show your working.

Cross-multiply: 5 × 24 = 120, 8 × 15 = 120. Equal, so yes. Or: 5/8 × 3/3 = 15/24.Yes, they are equivalent
4 Use cross-cancelling: $\frac{5}{6} \times \frac{3}{10}$

Cancel: 5 and 10 share 5, 6 and 3 share 3. (5÷5)/(6÷3) × (3÷3)/(10÷5) = 1/2 × 1/2 = 1/4.1/4

Complete in your workbook.

Quick Check · 5 questions

$\frac{3}{5} = \frac{?}{25}$

+10 XP

Simplify $\frac{18}{24}$ fully

+10 XP

Are $\frac{4}{7}$ and $\frac{6}{12}$ equivalent?

+10 XP

Simplify $\frac{64}{80}$ fully

+10 XP

$\frac{6}{7} \times \frac{14}{15}$ using cross-cancelling

+10 XP

Show Your Working · 3 questions

Show Your Working

10 marks total

ApplyMedium3 MARKS

Q6. (a) Find three fractions equivalent to $\frac{3}{4}$. (b) Explain why $\frac{3}{4} = \frac{3 \times 5}{4 \times 5}$.

Answer in your workbook.

ApplyMedium4 MARKS

Q7. Simplify fully: (a) $\frac{28}{42}$, (b) $\frac{45}{75}$, (c) $\frac{96}{120}$. Show the HCF used for each.

Answer in your workbook.

ReasonHard3 MARKS

Q8. A student says $\frac{1}{2} = \frac{1+2}{2+2} = \frac{3}{4}$. Explain why this is wrong, and show the correct way to find an equivalent fraction with denominator 4.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — 3 × 5 = 15, 5 × 5 = 25. So 3/5 = 15/25.

2. B — HCF(18, 24) = 6. 18 ÷ 6 = 3, 24 ÷ 6 = 4. So 3/4.

3. A — 4 × 12 = 48, 7 × 6 = 42. 48 ≠ 42, so not equivalent.

4. D — HCF(64, 80) = 16. 64 ÷ 16 = 4, 80 ÷ 16 = 5. So 4/5.

5. B — 6 and 15 share 3, 7 and 14 share 7. After cancelling: 2/1 × 2/5 = 4/5.

Show Your Working Model Answers

Q6 (3 marks): (a) Any three of: 6/8, 9/12, 12/16, 15/20, 30/40 [2 marks, 1 per correct fraction]. (b) Multiplying by 5/5 = 1, so value unchanged [1].

Q7 (4 marks): (a) HCF(28,42) = 14, so 28/42 = 2/3 [1.5]. (b) HCF(45,75) = 15, so 45/75 = 3/5 [1.5]. (c) HCF(96,120) = 24, so 96/120 = 4/5 [1].

Q8 (3 marks): Adding changes the value [1]. You must multiply or divide both parts by the same number [1]. Correct: 1/2 = (1×2)/(2×2) = 2/4 [1].

Stretch Challenge · +25 XP, +10 coins

The Fraction Chain

Start with $\frac{2}{3}$. Find a chain of 5 equivalent fractions where each one has a denominator exactly double the previous one. Then simplify the last fraction back to its lowest terms. What do you notice?

Reveal solution

$\frac{2}{3} \to \frac{4}{6} \to \frac{8}{12} \to \frac{16}{24} \to \frac{32}{48} \to \frac{64}{96}$. Simplify $\frac{64}{96}$: HCF = 32, so $\frac{64 \div 32}{96 \div 32} = \frac{2}{3}$. You get back to where you started! Equivalent fractions form an infinite loop.

Quick Review

Equivalent

Same value, different look

× or ÷ both

By the same number

Simplify

Divide by HCF

Lowest terms

HCF(num, den) = 1

Cross-multiply

a×d = b×c means equal

Cross-cancel

Only across ×, diagonally

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