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Ratio & Proportion · Higher

Compound interest & depreciation

Compound interest calculates interest on both the principal and previously earned interest. Depreciation is the decrease in value over time, calculated the same way but using a multiplier less than 1.

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Key facts to remember

  • 1Compound interest formula: A = P(1 + r/100)ⁿ, where P = principal, r = rate %, n = years.
  • 2For depreciation: A = P(1 − r/100)ⁿ.
  • 3The multiplier for r% compound interest is (1 + r/100); for r% depreciation is (1 − r/100).
  • 4Compound interest grows faster than simple interest over time.
  • 5After n years, apply the multiplier n times (or use the power n).
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Formulas

Compound interest
A = P(1 + r/100)ⁿ

A = final amount, P = principal, r = rate %, n = years

Depreciation
A = P(1 − r/100)ⁿ
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Worked examples

Example 1

£3000 is invested at 4% compound interest per year. Find the value after 5 years.

Working

  1. Multiplier = 1 + 4/100 = 1.04
  2. A = 3000 × 1.04⁵
  3. 1.04⁵ = 1.2166…
  4. A = 3000 × 1.2166 = £3649.96
Answer£3649.96 (to nearest penny)
Example 2

A car bought for £18,000 depreciates at 12% per year. Find its value after 3 years.

Working

  1. Multiplier = 1 − 0.12 = 0.88
  2. A = 18000 × 0.88³
  3. 0.88³ = 0.681472
  4. A = 18000 × 0.681472 = £12,266.50
Answer£12,266.50
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Common mistakes

✗Using simple interest (P × r × n / 100) instead of compound interest.
✗Using the wrong multiplier — an increase of 4% is × 1.04, not × 0.04.
✗Not applying the power n — applying the multiplier only once instead of n times.
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Exam tips

✓Write out A = P × (multiplier)ⁿ, identify each value, and use your calculator's power function.
✓Always check: compound interest should be slightly more than simple interest for the same rate and years.

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