Mathematics • Year 7 • Unit 1 • Lesson 7
Equivalent Fractions and Simplifying
Build the rule: whatever you do to the top, do to the bottom. Multiply or divide both parts by the same number to get an equivalent fraction; divide by the HCF to simplify.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Simplify 36/48 fully.
Step 1 — Find the HCF (highest common factor) 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 factors: 1, 2, 3, 4, 6, 12. HCF = 12.
Reason: the HCF is the largest number that divides both top and bottom exactly.
Step 2 — Divide both numerator and denominator by the HCF.
36 ÷ 12 = 3 48 ÷ 12 = 4
Reason: the golden rule — whatever you do to the top, do to the bottom. Dividing by 12/12 (which equals 1) doesn't change the value.
Step 3 — Write the new fraction.
36/48 = 3/4
Step 4 — Check it's fully simplified.
HCF(3, 4) = 1 → no more simplifying possible.
Answer: 3/4.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Simplify 18/24 fully.
Step 1 — Find common factors of 18 and 24:
Factors of 18: 1, 2, 3, _____, _____, 18.
Factors of 24: 1, 2, 3, _____, _____, _____, _____, 24.
HCF = _____.
Step 2 — Divide both top and bottom by the HCF:
18 ÷ _____ = _____ 24 ÷ _____ = _____
Step 3 — Write the simplified fraction:
18/24 = _____ / _____
Step 4 — Check: HCF of your final top and bottom? _____
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Find the missing number: 1/2 = ?/10. 1 mark
3.2 Find the missing number: 3/5 = ?/20. 1 mark
3.3 Simplify 6/8 fully. 1 mark
3.4 Simplify 10/15 fully. 1 mark
Standard — combine two ideas
3.5 Use the cross-multiplication rule to check: are 4/6 and 10/15 equivalent? Show every step. 2 marks
3.6 Simplify 24/36 fully by dividing by 2 repeatedly until you can't anymore, then by 3 if possible. 2 marks
Extension — push your thinking
3.7 Simplify 45/75 fully. (Hint: both end in 5 — what common factor does that suggest?) 3 marks
3.8 Use cross-cancellation to evaluate 3/8 × 4/9 in lowest terms. Show the cancelling step. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (18/24)
Step 1: Factors of 18 = 1, 2, 3, 6, 9, 18. Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24. HCF = 6.
Step 2: 18 ÷ 6 = 3, 24 ÷ 6 = 4.
Step 3: 18/24 = 3/4.
Step 4: HCF(3, 4) = 1, so fully simplified.
3.1 — 1/2 = ?/10
Bottom went from 2 to 10, so multiplied by 5. Do the same on top: 1 × 5 = 5. So 1/2 = 5/10.
3.2 — 3/5 = ?/20
Bottom 5 → 20 is × 4. Top: 3 × 4 = 12. So 3/5 = 12/20.
3.3 — Simplify 6/8
HCF(6, 8) = 2. 6 ÷ 2 = 3, 8 ÷ 2 = 4. So 6/8 = 3/4. Check HCF(3,4) = 1. ✓
3.4 — Simplify 10/15
HCF(10, 15) = 5. 10 ÷ 5 = 2, 15 ÷ 5 = 3. So 10/15 = 2/3. Check HCF(2,3) = 1. ✓
3.5 — Are 4/6 and 10/15 equivalent?
Cross-multiply: 4 × 15 = 60 and 6 × 10 = 60. Both equal 60, so yes, they are equivalent. (Both simplify to 2/3.)
3.6 — Simplify 24/36 step by step
Both even: 24 ÷ 2 = 12, 36 ÷ 2 = 18. So 12/18.
Still both even: 12 ÷ 2 = 6, 18 ÷ 2 = 9. So 6/9. (Now one is even, one is odd — can't divide by 2 again.)
Both divisible by 3: 6 ÷ 3 = 2, 9 ÷ 3 = 3. So 2/3. Check HCF(2,3) = 1. ✓
3.7 — Simplify 45/75
Both end in 5, so both divisible by 5. 45 ÷ 5 = 9, 75 ÷ 5 = 15. So 9/15.
Both divisible by 3: 9 ÷ 3 = 3, 15 ÷ 3 = 5. So 3/5. Check HCF(3, 5) = 1. ✓
(Faster: HCF(45, 75) = 15 directly. 45 ÷ 15 = 3, 75 ÷ 15 = 5.)
3.8 — 3/8 × 4/9 with cross-cancellation
Look diagonally: 3 (top-left) and 9 (bottom-right) share factor 3 → 3 ÷ 3 = 1, 9 ÷ 3 = 3. 4 (top-right) and 8 (bottom-left) share factor 4 → 4 ÷ 4 = 1, 8 ÷ 4 = 2.
After cancelling: 1/2 × 1/3 = 1/6. (Check without cancelling: 12/72 = 1/6. ✓)