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. ✓)