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%
❓ 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
✅ 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%
🎯 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).
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