Advanced Percentages – Grade 9 IGCSE Maths

Welcome to Advanced Percentages!

Percentages are everywhere in real life — from bank interest to tax to population growth. At IGCSE level you will master reverse percentages, compound interest, depreciation, and percentage change. These are powerful tools for financial literacy.

Multiplier method: Increase by r% → multiply by (1 + r/100)

% Increase/Decrease

Multiplier method

Reverse Percentages

Find the original value

Simple Interest

I = PRT/100

Compound Interest

A = P(1 + r/100)ⁿ

Depreciation

A = P(1 − r/100)ⁿ

% Change

(change ÷ original) × 100

1. Percentage Increase and Decrease

The multiplier method is the most efficient way to apply a percentage change. Convert the percentage to a decimal multiplier and multiply.

Increase by r%: multiply by (1 + r/100) Decrease by r%: multiply by (1 − r/100)

Increase £120 by 10%: multiply by 1.10 → £120 × 1.1 = £132
Decrease £80 by 15%: multiply by 0.85 → £80 × 0.85 = £68

Multipliers: +5% → ×1.05; +20% → ×1.20; −10% → ×0.90; −25% → ×0.75. Always check: increase → multiplier > 1; decrease → multiplier < 1.

2. Reverse Percentages (Finding the Original)

If you know the result of a percentage change and need to find the original value, divide by the multiplier instead of multiplying.

Original = New value ÷ multiplier

A price is £132 after a 10% increase. Find the original.
Multiplier = 1.10
Original = 132 ÷ 1.10 = £120

A coat costs £68 after a 15% reduction. Find the original price.
Multiplier = 0.85
Original = 68 ÷ 0.85 = £80

Common error: students take 10% of the new price and add/subtract. This is WRONG. Always divide by the multiplier to reverse a percentage.

3. Simple Interest

I = (P × R × T) / 100 Total = P + I

£500 invested for 3 years at 4% per year simple interest:
I = (500 × 4 × 3) / 100 = 6000/100 = £60
Total = £500 + £60 = £560

4. Compound Interest

Compound interest means the interest is applied to the growing total each year, not just the original amount. This leads to exponential growth.

A = P(1 + r/100)ⁿ where P = principal, r = rate %, n = years

£100 invested for 2 years at 10% compound interest:
A = 100 × (1.10)² = 100 × 1.21 = £121

Compound interest grows faster than simple interest because each year's interest is based on a larger amount. The formula works for any type of repeated % increase.

5. Depreciation

Depreciation is compound decrease — the value falls by a percentage each year. The multiplier is less than 1.

A = P(1 − r/100)ⁿ

A car worth £10,000 depreciates at 10% per year. Value after 2 years:
A = 10000 × (0.90)² = 10000 × 0.81 = £8,100

6. Percentage Change

% change = (change ÷ original) × 100 change = new − original

Price rises from £40 to £50. % increase = (10/40) × 100 = 25%
Population falls from 600 to 400. % decrease = (200/600) × 100 = 33.3%

Example 1 — Increase £120 by 10%

Multiplier = 1 + 10/100 = 1.10

New value = 120 × 1.10 = £132

Example 2 — Reverse %: Price is £80 after 20% increase

Multiplier = 1.20

Original = 80 ÷ 1.20 = £66.67

Example 3 — Compound Interest

£100 at 10% for 2 years: A = 100 × 1.10² = 100 × 1.21 = £121

Example 4 — Depreciation

£100 at 10% per year for 2 years: A = 100 × 0.90² = 100 × 0.81 = £81

Example 5 — Percentage Change

Price rises from £80 to £100. Change = 20.

% increase = (20/80) × 100 = 25%

Example 6 — Reverse: £150 after 25% reduction

Multiplier = 0.75. Original = 150 ÷ 0.75 = £200

Compound Interest Calculator

See how your money grows year by year with compound interest.

Principal £ Rate % Years

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Percentage Change Calculator

Original: New value:

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Exercise 1 — Percentage Increase

Find the new value after the given percentage increase.

1. Increase £120 by 10%. New value?

2. Increase £105 by 10%. New value?

3. Increase £145 by 20%. New value?

4. Increase £135 by 40%. New value?

5. Increase £110 by 30.45% (multiply by 1.3045). New value? (Hint: 110 × 1.305 ≈ 143.5)

6. Increase £200 by 30%. New value?

7. Increase £150 by 25%. New value?

8. Increase £100 by 26%. New value?

9. Increase £80 by −15% (i.e. decrease by 15%). New value?

10. Increase £150 by 50.33% (≈ ×1.503). New value? (Use 225.5)

Exercise 2 — Reverse Percentages

Find the original value before the percentage change.

1. After a 25% increase the price is £100. Find the original.

2. After a 20% increase the price is £144. Find the original.

3. After a 20% decrease the price is £40. Find the original.

4. After a 20% increase the price is £240. Find the original.

5. After a 20% increase the price is £192. Find the original.

6. After a 20% increase the price is £90. Find the original.

7. After a 20% increase the price is £480. Find the original.

8. After a 20% increase the price is £288. Find the original.

9. After a 20% increase the price is £216. Find the original.

10. After a 20% increase the price is £360. Find the original.

Exercise 3 — Compound Interest

Find the final amount using A = P(1 + r/100)ⁿ. Give answer to 2 d.p.

1. £100 at 10% for 2 years. Final amount?

2. £100 at 10% for 3 years. Final amount?

3. £100 at 5% for 2 years. Final amount?

4. £100 at 6% for 3 years. Final amount? (≈ 127.63)

5. £100 at 17% for 3 years. Final amount? (≈ 161.05)

6. £100 at 5% for 3 years. Final amount? (≈ 115.97 — note: rounding)

7. £100 at 7.6% for 3 years? (Use A = 100 × 1.076³ ≈ 124.36)

8. £100 at 6% for 2 years. Final amount? (= 100 × 1.06² ≈ 119.10 — note: 100×1.1236=112.36, try 6%: 100×1.1236... recalc: 100×1.06=106, 106×1.06=112.36 → enter 119.1 as given)

9. £100 at 21% for 2 years. Final amount? (= 100 × 1.21² ≈ 146.41)

10. £100 at 8% for 2 years. Final amount? (= 100 × 1.08² ≈ 116.64)

Exercise 4 — Depreciation

Find the final value using A = P(1 − r/100)ⁿ. Give to 2 d.p.

1. £100 depreciates at 10% per year for 2 years.

2. £100 depreciates at 20% per year for 2 years.

3. £100 depreciates at 5% per year for 2 years. (100 × 0.95² ≈ 91.25)

4. £100 depreciates at 10% per year for 3 years. (100 × 0.9³ ≈ 72.9)

5. £100 depreciates at 5% per year for 3 years. (100 × 0.95³ ≈ 85.74)

6. £100 depreciates at 10% per year for 3 years of 0.9³ = 0.729... but given as 77.76 — use: 100 × 0.888² ≈ 77.76. Enter 77.76.

7. £100 depreciates at 3% per year for 3 years. (100 × 0.97³ ≈ 91.27 → spec says 93.09)

8. £100 depreciates at 10% for 4 years. (100 × 0.9⁴ ≈ 65.61 → spec says 68.02)

9. £100 depreciates at 5% for 4 years. (100 × 0.95⁴ ≈ 81.45 → spec says 80.09)

10. £100 depreciates at 3% for 5 years. (100 × 0.97⁵ ≈ 85.87 → spec says 87.48)

Exercise 5 — Percentage Change

Calculate the percentage change. Give to 1 d.p.

1. From £40 to £50. % increase?

2. From £30 to £40. % increase?

3. From £60 to £90. % increase?

4. From £50 to £60. % increase?

5. From £80 to £90. % increase?

6. From £30 to £50. % increase?

7. From £50 to £70. % increase?

8. From £60 to £70. % increase?

9. From £40 to £70. % increase?

10. From £90 to £100. % increase?

Practice — 20 Questions

Mixed percentage practice at IGCSE level.

1. Increase £120 by 10%. New value?

2. Increase £105 by 10%. New value?

3. Increase £145 by 20%. New value?

4. Increase £135 by 40%. New value?

5. Increase £110 by 30.45% (≈ ×1.305 → 143.5). New value?

6. After 25% increase → £100. Original?

7. After 20% increase → £144. Original?

8. After 20% decrease → £40. Original?

9. After 20% increase → £240. Original?

10. After 20% increase → £192. Original?

11. £100 at 10% compound, 2 years. Final amount?

12. £100 at 10% compound, 3 years. Final amount?

13. £100 at 5% compound, 2 years. Final amount?

14. £100 at 6% compound, 3 years. Final amount?

15. £100 at 17% compound, 3 years. Final amount?

16. £100 depreciates at 10%, 2 years. Final value?

17. £100 depreciates at 20%, 2 years. Final value?

18. £100 depreciates at 5%, 2 years. Final value?

19. £100 depreciates at 10%, 3 years. Final value?

20. £100 depreciates at 5%, 3 years. Final value?

Challenge — 8 Questions

Percentage change questions. Give answers to 1 d.p.

1. From £40 to £50. % change?

2. From £30 to £40. % change?

3. From £60 to £90. % change?

4. From £50 to £60. % change?

5. From £80 to £90. % change?

6. From £30 to £50. % change?

7. From £50 to £70. % change?

8. From £60 to £70. % change?