N12: Fractions & Percentages as Operators
Interpret fractions and percentages as operators
📋 Key Concepts
Operator: In mathematics, "of" means multiply. So "3/4 of 20" means 3/4 × 20, and "25% of 80" means 25% × 80.
Key Conversions:
1/2 = 50%
1/4 = 25%
3/4 = 75%
1/5 = 20%
1/10 = 10%
1/2 = 50%
1/4 = 25%
3/4 = 75%
1/5 = 20%
1/10 = 10%
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
📝 Finding a Fraction of an Amount
Method: Divide by the denominator, then multiply by the numerator.
Example 1
Find 3/4 of £48
Solution:
Step 1: £48 ÷ 4 = £12
Step 2: £12 × 3 = £36
Example 2
Find 2/5 of 35 kg
Solution:
Step 1: 35 ÷ 5 = 7
Step 2: 7 × 2 = 14 kg
📝 Finding a Percentage of an Amount
Method 1: Convert percentage to fraction and calculate.
Method 2: Use calculator: multiply by percentage as decimal.
Method 2: Use calculator: multiply by percentage as decimal.
Example 3
Find 30% of £80
Solution (non-calculator):
10% of £80 = £8
30% = 3 × 10% = 3 × £8 = £24
Example 4
Find 15% of £240
Solution:
10% of £240 = £24
5% of £240 = £12
15% = £24 + £12 = £36
Example 5
Find 17% of £600
Solution (calculator):
17% = 0.17
0.17 × 600 = £102
📝 Percentage Increase and Decrease
Increase: Add the percentage to 100%, then multiply.
Decrease: Subtract the percentage from 100%, then multiply.
Decrease: Subtract the percentage from 100%, then multiply.
Example 6
Increase £80 by 25%
Solution:
100% + 25% = 125% = 1.25
£80 × 1.25 = £100
Example 7
Decrease £150 by 20%
Solution:
100% - 20% = 80% = 0.8
£150 × 0.8 = £120
📝 Multipliers
Common Multipliers:
Increase by 10%: × 1.1
Increase by 25%: × 1.25
Decrease by 20%: × 0.8
Decrease by 30%: × 0.7
Increase by 10%: × 1.1
Increase by 25%: × 1.25
Decrease by 20%: × 0.8
Decrease by 30%: × 0.7
Example 8
A TV costs £400. In a sale it has 35% off. Find the sale price.
Solution:
Multiplier for 35% decrease: 1 - 0.35 = 0.65
£400 × 0.65 = £260
📝 Expressing One Quantity as a Percentage of Another
Method: Create a fraction, then multiply by 100.
Example 9
Express £18 as a percentage of £72
Solution:
Fraction = 18/72 = 1/4
1/4 × 100 = 25%
Example 10
In a test, Sarah scored 42 out of 60. What is her percentage?
Solution:
42/60 × 100 = 70%
🧮 Worked Examples — Exam Style
Percentage change questions chain two operations — and the trap is assuming +20% then −10% returns near the start. Multipliers do the bookkeeping.
A jumper costs £45. The price increases by 20%, then the new price decreases by 10%. What is the final price?
Increase: ×1.20 → £45 × 1.20 = £54
Decrease: ×0.90 → £54 × 0.90 = £48.60
Not £45 × 1.10 — the 10% comes off the NEW price.
M1 for 45 × 1.2, M1 for 54 × 0.9, A1 for £48.60. The net effect is ×1.08 (a rise), not ×1.10.
The value of a car rises from £80 to £92. Write the percentage increase.
Change = 92 − 80 = £12. Percentage = 12 ÷ 80 × 100 = 15%
The rule: change ÷ ORIGINAL × 100 — divide by where you started.
M1 for 12/80, A1 for 15%. Dividing by 92 instead of 80 is the standard slip.
Exam wording to expect: "Find …% of …", "Increase/decrease by …%", "Express … as a percentage of …".
❓ Practice Questions
Q1: Find 2/3 of £54
Q2: Find 35% of 240
Q3: Increase £180 by 15%
Q4: Decrease £250 by 30%
Q5: Express 45 as a percentage of 150
Q6: Find 12.5% of 640
Q7: Express 18/45 as a percentage
✅ Answers
- 54 ÷ 3 = 18, 18 × 2 = £36
- 35% = 0.35, 0.35 × 240 = 84
- 115% = 1.15, £180 × 1.15 = £207
- 70% = 0.7, £250 × 0.7 = £175
- 45/150 × 100 = 30%
🧮 Worked Examples — Harder Exam Style
A phone priced at £240 is discounted by 15% in a sale. The sale price then rises by 15%. Explain why the final price is not £240.
Sale price: 240 × 0.85 = £204
Rise: 204 × 1.15 = £234.60
Why not £240: the 15% rise applies to the SMALLER sale price (£204), so it adds only £30.60, while the discount removed £36. The percentage is of different bases.
M1 for 0.85 and 1.15 multipliers, A1 for £234.60, B1 for the explanation. The explanation line is explicitly asked — 'the percentages are of different amounts' scores.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 54 ÷ 3 = 18, then 18 × 2 = £36
Q2. 35% = 0.35 → 0.35 × 240 = 84
Q3. Increase by 15% → multiplier 1.15 → 180 × 1.15 = £207. One multiplication replaces the find-10%-and-add method.
Q4. Decrease by 30% → multiplier 0.70 → 250 × 0.7 = £175
Q5. 45 ÷ 150 × 100 = 30%
Q6. 12.5% = 1/8 → 640 ÷ 8 = 80
Q7. 18/45 = 2/5 (÷9) = 40/100 → 40%
📌 Key Facts to Memorise
- Multiplier table: +25% ×1.25 · +10% ×1.1 · +5% ×1.05 · −20% ×0.8 · −15% ×0.85
- Percentage change = change ÷ ORIGINAL × 100
- Fraction of amount: ÷ bottom, × top
- Common fraction-percentages: 50% = 1/2 · 25% = 1/4 · 12.5% = 1/8 · 10% = 1/10
- Chained changes multiply: +20% then −10% = ×1.2 × 0.9 = ×1.08 (net +8%)
🗺️ In the Exam
Percentages of amounts: 3–5 marks per paper — VAT, interest, discounts, depreciation. The chained-change version and reverse-percentage appear on Papers 2/3 (calculator).
Links to: fractions of amounts (R3), interest and growth (R9), reverse percentages (Higher R9).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- "Of" always means multiply
- Build up from easy percentages: 50%, 10%, 5%, 1%
- For percentage change: multiplier = (100 ± %)/100
- Check: percentage should be between 0% and 100% for "part of whole"
- Memorise common fraction/percentage equivalents
🧠 Problem-Solving Strategies
Problem-Solving
For percentage problems, always identify the original value (100%). For successive percentage changes, use multipliers — don't just add the percentages. Remember that a 10% increase followed by a 10% decrease does NOT return to the original value. Build up from simple percentages (50%, 10%, 5%, 1%) for non-calculator methods.
Multi-Step Problem
A jacket costs £120. It is reduced by 15% in a sale, then the sale price is reduced by a further 20%. Find the final price and the overall percentage reduction from the original price.
Solution: After 15% off: £120 × 0.85 = £102. After further 20% off: £102 × 0.80 = £81.60. Overall reduction = £120 − £81.60 = £38.40. Overall % = 38.40/120 × 100 = 32%. Note: this is NOT 15% + 20% = 35%.
⚠️ Common Errors
Watch Out!
1. Wrong: A 20% increase then 20% decrease gets back to the original Correct: 100 × 1.2 × 0.8 = 96 — you end up at 96% of the original
2. Wrong: 3/4 of 60 = 60 ÷ 4 × 3 = 45, but 3/4 of 8 = 8 ÷ 3 × 4 Correct: Always divide by denominator first: 8 ÷ 4 × 3 = 6
3. Wrong: Percentage increase from 40 to 50 is 10% (just subtracting) Correct: Increase = 10. Percentage = 10/40 × 100 = 25% (divide by the ORIGINAL)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A house was bought for £180,000. After 3 years its value increased by 12%. It was then sold, and the seller paid 1.5% estate agent fees on the sale price. (a) Find the sale price. (b) Find the estate agent fees. (c) Calculate the actual percentage profit the seller made on the original purchase price after fees are deducted.
(a) Sale price = £180,000 × 1.12 = £201,600.
(b) Fees = 1.5% of £201,600 = £201,600 × 0.015 = £3,024.
(c) Profit after fees = £201,600 − £3,024 − £180,000 = £18,576. Percentage profit = 18,576/180,000 × 100 = 10.32%.
Mark scheme: 2 marks for (a), 1 mark for (b), 3 marks for (c) including correct profit calculation and percentage
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Shop A has a coat for £160 with "1/3 off" in the sale. Shop B has the same coat for £180 with "40% off".
(a) Which shop offers the cheaper sale price?
(b) How much money would you save by choosing the cheaper option?
(c) Shop A then offers an extra 10% off the already discounted price. Does this make Shop A definitely cheaper than Shop B now? Show working to justify your answer.
Answers: (a) Shop A: £160 × 2/3 = £106.67. Shop B: £180 × 0.60 = £108. Shop A is cheaper. (b) Save £108 − £106.67 = £1.33. (c) Extra 10% off: £106.67 × 0.90 = £96.00. Shop B is still £108. Yes, Shop A is now definitely cheaper (£96 vs £108).
📝 Exam Questions by Topic
🎬 Video Resources
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