Percentages: The Full Picture

Grade 7 — Cambridge Lower Secondary Stage 7, Unit 10. Master multipliers, reverse percentages, percentage change, and compound interest.

The Multiplier Machine

Every percentage increase or decrease can be written as a single multiplier. Watch the machine cycle through examples:

Increase by 20% — multiplier is 1.20
Original Value £50
×
Multiplier 1.20
=
New Value £60
1 / 8

1. Percentage Increase & Decrease — Multipliers

To increase by p%, multiply by (1 + p/100). To decrease by p%, multiply by (1 − p/100).

New Value = Original × Multiplier

Examples: +15% → ×1.15  |  −15% → ×0.85  |  +7.5% → ×1.075  |  −3% → ×0.97

2. Reverse Percentages — Finding the Original

If you know the final value after a percentage change, divide by the multiplier to get back to the original.

Original = Final Value ÷ Multiplier

Example: A coat costs £78 after a 30% increase. Original = £78 ÷ 1.30 = £60

Key mistake to avoid: Do NOT just subtract 30% from £78. That gives the wrong answer!

3. Percentage Change Formula

% Change = (Change ÷ Original) × 100

Where Change = New Value − Original Value (positive = increase, negative = decrease).

Example: Price rises from £40 to £52. Change = 12. % change = (12 ÷ 40) × 100 = 30%

4. Compound Interest & Compound Percentage Change

Compound means the percentage is applied again and again on the new amount each time.

Final Amount = Principal × (Multiplier)n

Where n = number of periods (years, etc.).

Example: £1000 at 5% compound interest for 3 years = £1000 × 1.05³ = £1000 × 1.157625 = £1157.63

This is different from simple interest which uses the same original amount each time.

5. Real-World Percentage Problems

  • VAT (20% in UK): Price with VAT = Price × 1.20  |  Price without VAT = Price with VAT ÷ 1.20
  • Discount: Sale price = Original × (1 − discount%/100)
  • Profit %: (Profit ÷ Cost Price) × 100
  • Loss %: (Loss ÷ Cost Price) × 100

6. Expressing One Quantity as a Percentage of Another

Percentage = (Part ÷ Whole) × 100

Example: 36 out of 48 students passed. Percentage = (36 ÷ 48) × 100 = 75%

Both quantities must be in the same units before dividing!

Worked Examples

Study these carefully — they show the full method for each type.

Example 1: Percentage Increase Using a Multiplier

Question: Increase £45 by 20%.

Step 1 — Find the multiplier: 20% increase → 1 + 0.20 = 1.20
Step 2 — Multiply: £45 × 1.20 = £54

Check: 20% of £45 = £9, so £45 + £9 = £54. ✓

Answer: £54

Example 2: Percentage Decrease Using a Multiplier

Question: Decrease 80 kg by 15%.

Step 1 — Find the multiplier: 15% decrease → 1 − 0.15 = 0.85
Step 2 — Multiply: 80 × 0.85 = 68 kg

Check: 15% of 80 = 12, so 80 − 12 = 68 kg. ✓

Answer: 68 kg

Example 3: Reverse Percentage (Finding the Original Price)

Question: A jacket costs £91 after a 30% increase. What was the original price?

Step 1 — Identify the multiplier: 30% increase → multiplier = 1.30
Step 2 — The final price IS the original × 1.30. So: Original = £91 ÷ 1.30
Step 3 — Calculate: £91 ÷ 1.30 = £70

Verify: £70 × 1.30 = £91. ✓

Answer: £70

Common Error: Do NOT do £91 × 0.70 — that gives £63.70 which is WRONG.

Example 4: Compound Interest Over 3 Years

Question: £2000 is invested at 4% compound interest per year. Find the value after 3 years.

Step 1 — Multiplier for 4% increase = 1.04
Step 2 — Apply for 3 years: £2000 × 1.04³
Step 3 — Calculate 1.04³: 1.04 × 1.04 = 1.0816; 1.0816 × 1.04 = 1.124864
Step 4 — £2000 × 1.124864 = £2249.73 (to nearest penny)

Compare with simple interest: Simple would give £2000 + 3 × £80 = £2240. Compound gives more because interest is earned on interest.

Answer: £2249.73

Percentage Change Calculator

Exercise 1: Multiplier Match

Type the multiplier for each percentage change. Use decimals (e.g. 1.35 for +35%, 0.72 for −28%).

Exercise 2: Increase & Decrease

Use the multiplier method to find the new value. Round to 2 decimal places where needed.

Exercise 3: Reverse Percentages

Find the original value before the percentage change. Round to 2 decimal places.

Hint: Reverse % means DIVIDE by the multiplier.
If a price increased by 25% to reach the final value, the multiplier was 1.25.
Original = Final Value ÷ 1.25

Exercise 4: Percentage Change

Use the formula: % Change = (Change ÷ Original) × 100. Give answers to 1 decimal place.

Exercise 5: Compound Interest

Use: Final = Principal × (Multiplier)n. Round to 2 decimal places (nearest penny).

Method:
1. Write the multiplier: e.g. 3% → 1.03
2. Raise it to the power n: 1.03² = 1.0609; 1.03³ = 1.092727
3. Multiply by Principal
Example: £500 at 6% for 2 years = £500 × 1.06² = £500 × 1.1236 = £561.80

Exercise 6: Real World Scenarios

Click the correct answer. Immediate feedback with explanation!

Practice Questions

Work these out on paper. Use the multiplier method where possible.

  1. Increase £120 by 25%.
  2. Decrease 200 g by 35%.
  3. Increase 64 km by 12.5%.
  4. Decrease £85 by 40%.
  5. A pair of trainers cost £60 before VAT (20%). What is the price including VAT?
  6. A TV costs £336 in a 30% off sale. What was the original price?
  7. A concert ticket costs £46 after a 15% increase. What was the original price?
  8. After a 20% pay rise, a worker earns £2160 per month. What was their original monthly pay?
  9. A car was bought for £8500 and sold for £6800. Calculate the percentage loss.
  10. A house bought for £180,000 is sold for £216,000. Calculate the percentage profit.
  11. A population grows from 45,000 to 48,600. Calculate the percentage increase.
  12. Express 48 out of 64 as a percentage.
  13. In a class of 30 students, 18 are girls. What percentage are boys?
  14. £500 is invested at 3% compound interest per year. Find the value after 2 years.
  15. £1200 is invested at 5% compound interest per year. Find the value after 3 years.
  16. A bacteria colony of 800 doubles (increases by 100%) each hour. How many after 3 hours?
  17. A car depreciates (decreases) by 12% per year. It is worth £15,000 now. Find its value after 2 years.
  18. A shop buys a jacket for £40 and sells it for £58. Find the percentage profit.
  19. After a 4% increase and then a 4% decrease, is the final price the same as the original? Explain.
  20. A price is increased by 60% and then decreased by 25%. What is the overall percentage change?
  1. £120 × 1.25 = £150
  2. 200 × 0.65 = 130 g
  3. 64 × 1.125 = 72 km
  4. £85 × 0.60 = £51
  5. £60 × 1.20 = £72
  6. £336 ÷ 0.70 = £480
  7. £46 ÷ 1.15 = £40
  8. £2160 ÷ 1.20 = £1800
  9. Loss = £1700; (1700 ÷ 8500) × 100 = 20% loss
  10. Profit = £36,000; (36000 ÷ 180000) × 100 = 20% profit
  11. Increase = 3600; (3600 ÷ 45000) × 100 = 8% increase
  12. (48 ÷ 64) × 100 = 75%
  13. Boys = 12; (12 ÷ 30) × 100 = 40%
  14. £500 × 1.03² = £500 × 1.0609 = £530.45
  15. £1200 × 1.05³ = £1200 × 1.157625 = £1389.15
  16. 800 × 2³ = 800 × 8 = 6400 bacteria
  17. £15,000 × 0.88² = £15,000 × 0.7744 = £11,616
  18. Profit = £18; (18 ÷ 40) × 100 = 45%
  19. No. Original = £100; after +4% = £104; after −4% of £104 = £104 × 0.96 = £99.84 (slightly less)
  20. Multiplier = 1.60 × 0.75 = 1.20; overall = +20% increase

Challenge: Hard Word Problems

Multi-step problems. Show all workings. Think carefully!

  1. A laptop is advertised as "25% off, now £360". A different shop sells the same laptop with 30% off the original price. If the original price is the same in both shops, which deal saves more money, and by how much?
  2. Marcus invests £3000 at 4.5% compound interest per year. Sarah invests £3000 at 4% compound interest per year. How much more does Marcus have after 3 years? (Give your answer to the nearest penny.)
  3. A shop increases all prices by 20% for summer, then reduces them by 20% in the January sale. A customer says "the prices are back to normal." Is the customer correct? Show working to justify your answer.
  4. A dress is priced at £84 including 20% VAT. Sarah has a coupon giving 15% off the VAT-inclusive price. How much does Sarah pay? What is the percentage saving on the original pre-VAT price?
  5. Tom bought shares for £2500. They increased in value by 18% in the first year, then decreased by 10% in the second year. What is the value of his shares after two years? What is the overall percentage change from the original price?
  6. A town's population was 24,000 in 2020. It grew by 5% in 2021 and by 8% in 2022. What was the population at the end of 2022? By what single percentage did the population grow over the two years? (Give to 1 d.p.)
  7. A car dealer buys a car for £6000 and wants to make a 35% profit. At what price should he sell it? He actually sells it for £7500. What percentage profit does he actually make? (Give to 1 d.p.)
  8. Emma scored 68 out of 80 on a maths test and 54 out of 65 on a science test. In which subject did she perform better as a percentage? What is the difference in percentage scores? (Give to 1 d.p.)
  9. A house was worth £200,000 in 2015. Its value increased by 6% per year for 4 years. What was it worth in 2019 (to the nearest pound)? How much more is this than if it had simply increased by 24% from the original price?
  10. Two investment accounts both start with £5000. Account A earns 6% compound interest annually for 5 years. Account B earns 30% simple interest over 5 years (i.e. 6% of the original each year). Which account gives more money after 5 years, and by how much? (Give to the nearest penny.)
  1. Original price = £360 ÷ 0.75 = £480.
    Shop 1 saving = 25% of £480 = £120 → price £360.
    Shop 2 saving = 30% of £480 = £144 → price £336.
    Shop 2 saves more by £144 − £120 = £24.
  2. Marcus: £3000 × 1.045³ = £3000 × 1.141166... = £3423.50
    Sarah: £3000 × 1.04³ = £3000 × 1.124864 = £3374.59
    Marcus has £3423.50 − £3374.59 = £48.91 more.
  3. After +20%: £100 × 1.20 = £120.
    After −20% of £120: £120 × 0.80 = £96.
    Final price is £96, not £100. The customer is wrong — the price is 4% lower than original.
    The multiplier is 1.20 × 0.80 = 0.96, meaning a 4% overall decrease.
  4. Pre-VAT price = £84 ÷ 1.20 = £70.
    Sarah pays: £84 × 0.85 = £71.40.
    Saving compared to pre-VAT price: £70 − £71.40 = −£1.40 (she pays MORE than pre-VAT).
    Actually comparing to the £84 price: saving = £84 − £71.40 = £12.60.
    Sarah pays £71.40, saving £12.60 (15%) on the £84 price.
    As % of pre-VAT £70: (£84 − £71.40) ÷ £70 × 100 = 18% above pre-VAT price.
  5. After year 1 (+18%): £2500 × 1.18 = £2950.
    After year 2 (−10%): £2950 × 0.90 = £2655.
    Overall change: £2655 − £2500 = £155 increase.
    Value = £2655. Overall % change = (155 ÷ 2500) × 100 = +6.2% increase.
  6. After 2021 (+5%): 24,000 × 1.05 = 25,200.
    After 2022 (+8%): 25,200 × 1.08 = 27,216.
    Population = 27,216.
    Overall multiplier = 1.05 × 1.08 = 1.134 → 13.4% increase overall.
  7. Target price (35% profit): £6000 × 1.35 = £8100.
    Actual selling price: £7500.
    Actual profit: £7500 − £6000 = £1500.
    Actual % profit = (1500 ÷ 6000) × 100 = 25%.
    (He priced for 35% profit but only achieved 25%.)
  8. Maths: (68 ÷ 80) × 100 = 85%.
    Science: (54 ÷ 65) × 100 = 83.1% (1 d.p.).
    Emma did better in Maths by 85% − 83.1% = 1.9 percentage points.
  9. Compound: £200,000 × 1.06⁴ = £200,000 × 1.26247696 = £252,495 (nearest £).
    Simple 24% increase: £200,000 × 1.24 = £248,000.
    Compound value = £252,495. Difference = £252,495 − £248,000 = £4,495 more with compound.
  10. Account A (compound): £5000 × 1.06⁵ = £5000 × 1.3382255776 = £6691.13.
    Account B (simple 30% total): £5000 × 1.30 = £6500.00.
    Account A gives more: £6691.13 − £6500.00 = £191.13 more.
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