Grade 11 Finance, Growth and Decay

Grade 11 finance: straight-line and reducing-balance depreciation, nominal and effective interest rates, and timelines with changing rates, with worked…

Finance, growth and decay is about 15 of the 150 marks in Grade 11 Paper 1. It builds on the Grade 10 simple and compound interest formulas and adds decay: depreciation on the straight-line method, A = P(1 − in), and on the reducing-balance method, A = P(1 − i)ⁿ.

You also convert between nominal and effective interest rates with 1 + i_eff = (1 + i_nom/m)ᵐ, and use timelines to handle problems where the interest rate changes or money is deposited or withdrawn part of the way through. Grade 12 adds annuities and loans.

Key concepts

  • Simple decay (straight-line depreciation): A = P(1 − in)
  • Compound decay (reducing balance): A = P(1 − i)ⁿ
  • Nominal versus effective annual interest rate
  • 1 + i_eff = (1 + i_nom/m)ᵐ, where m is the number of compounding periods per year
  • Timelines for changing interest rates, deposits and withdrawals
  • Finding the rate i or the time n from a given value

Worked examples

Equipment worth R50 000 depreciates at 15% p.a. on the reducing balance. Find its value after 2 years.

A = P(1 − i)ⁿ = 50 000(0.85)² = R36 125. On the straight-line method it would be 50 000(1 − 0.15 × 2) = R35 000.

Find the effective annual rate of 12% p.a. compounded monthly.

1 + i_eff = (1 + 0.12/12)¹² = (1.01)¹² = 1.1268. So the effective rate is 12.68% p.a.

R10 000 is invested at 8% p.a. compounded quarterly for 3 years. Find the final amount.

i = 0.08/4 = 0.02 per quarter and n = 3 × 4 = 12 quarters. A = 10 000(1.02)¹² = R12 682.42.

R5 000 is invested at 6% p.a. compounded annually for 2 years, then at 9% p.a. for 3 more years. Find the final amount.

Use a timeline: A = 5 000(1.06)²(1.09)³ = 5 000 × 1.1236 × 1.295029 = R7 275.47.

Common mistakes

  • Using (1 + i) instead of (1 − i) for depreciation.
  • Forgetting to divide the annual rate by m and multiply n by m when interest is compounded monthly or quarterly.
  • Adding interest rates instead of multiplying growth factors when the rate changes.
  • Rounding the effective rate too early.

Exam tips

  • Draw a timeline for any question with more than one rate or a deposit part of the way through.
  • Write P, i, n (and m) before you substitute.
  • Round only the final answer, to the nearest cent.

Frequently asked questions

What is the difference between straight-line and reducing-balance depreciation?

Straight-line depreciation removes the same amount each year (a percentage of the original value). Reducing-balance depreciation removes a percentage of the current value, so the yearly loss gets smaller.

Why is the effective rate higher than the nominal rate?

Because interest is added more than once a year, and later interest is earned on earlier interest. The effective rate is the true yearly growth.

Are annuities in Grade 11?

No. Future and present value annuities are Grade 12 work.

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