Fractions and percentages as operators
A fraction or percentage in front of "of" is just a multiplication waiting to happen. Treat ¾ of … and 75% of … as the operator that multiplies whatever follows.
"Of" means "×"
- ¾ of 80 = ¾ × 80 = 60.
- 35% of 200 = 0.35 × 200 = 70.
- 1/3 of 9 = 1/3 × 9 = 3.
Percentage as a decimal multiplier
Convert percent to decimal by dividing by 100, then multiply.
- 12% of £45 = 0.12 × £45 = £5.40.
- 7.5% of 200 g = 0.075 × 200 = 15 g.
- 130% of 50 = 1.3 × 50 = 65.
Non-calculator percentage tricks
Build a percentage from blocks of 10%, 5% and 1%.
- 10% = ÷ 10.
- 1% = ÷ 100.
- 5% = half of 10%.
- 50% = ÷ 2.
- 25% = ÷ 4.
Worked example: 35% of £80 (no calculator).
- 10% = £8 → 30% = £24.
- 5% = £4 → 35% = £24 + £4 = £28.
Fraction operators on quantities
For "fraction of X":
- Divide X by the denominator (gives one share).
- Multiply by the numerator.
Example: 5/8 of £64 = (£64 ÷ 8) × 5 = £8 × 5 = £40.
Combining operators
If you apply two percentage changes in sequence, multiply the multipliers, NOT add the percents.
Worked example: a price rises 10%, then falls 10%. Final?
- ×1.10 then ×0.90 = ×0.99 → 1% lower than the start, NOT the original price.
This is percentage change territory and is examined heavily.
Increase / decrease by a percentage
To increase by P%, multiply by 1 + P/100. To decrease by P%, multiply by 1 − P/100.
- Increase 240 by 15%: 240 × 1.15 = 276.
- Decrease 240 by 15%: 240 × 0.85 = 204.
Reverse percentage problems
If a quantity has been multiplied by some factor, divide by that factor to undo it. This is a Higher-tier favourite.
Worked example: a TV costs £288 after a 20% discount. What was the original price?
- After discount = 80% of original = original × 0.80.
- Original = £288 ÷ 0.80 = £360.
⚠ Common error: students compute 20% of £288 = £57.60 and add it back. That's wrong — the 20% must be of the original price, not the discounted one.
Repeated percentage change (compound)
Each year a quantity is multiplied by the same factor.
Worked example: £5,000 invested at 4% compound interest for 3 years.
- Multiplier = 1.04.
- After 3 years: 5000 × 1.04³ = 5000 × 1.124864 = £5,624.32.
Percentage of a percentage
15% of 40% of 200:
- 0.4 × 200 = 80.
- 0.15 × 80 = 12.
Or chain in one go: 0.15 × 0.40 × 200 = 12.
⚠Common mistakes— Common mistakes (examiner traps)
- Treating a percentage rise of P% then a fall of P% as no change. It always leaves you with less, because the second multiplier is applied to the increased amount.
- Reverse percentages handled forward. £288 / 0.80 vs (£288 + 20% of £288).
- Using ÷100 twice when converting a percent to a decimal and then writing it as a fraction. Pick ONE form.
- Multiplying by 1.5 for a 5% increase instead of 1.05.
- Forgetting to round money to 2 d.p. in final answers.
➜Try this— Quick check
(a) 30% of 480 = ? (b) ⅗ of £45 = ? (c) Increase 250 by 12%. (d) The price after a 25% reduction is £75. What was the original?
Answers: (a) 144; (b) £27; (c) 280; (d) £75/0.75 = £100.
AI-generated · claude-opus-4-7 · v3-deep-number