1. **State the problem:** You owe R 500 000 and want to find which of the three payment plans pay closest to this amount without overpaying, considering different interest rates and compounding periods. 2. **Analyze Payment Plan 1:** - Immediate payment: R 100 000 (no interest) - Payment at end of 3 years: R 400 000 - Interest rate: 10% compounded annually (assumed) Calculate the present value (PV) of R 400 000 at 10% for 3 years: $$PV = \frac{400000}{(1+0.10)^3} = \frac{400000}{1.331} \approx 300526.32$$ Total PV for Plan 1 = 100000 + 300526.32 = 400526.32 3. **Analyze Payment Plan 2:** - Monthly payment: R 20 000 - Number of months: 3 years × 12 = 36 - Interest rate: 11% pa compounded monthly, so monthly interest rate: $$i = \frac{0.11}{12} = 0.0091667$$ Calculate the present value of the annuity (payments at the end of each month): $$PV = P \times \frac{1-(1+i)^{-n}}{i}$$ $$PV = 20000 \times \frac{1-(1+0.0091667)^{-36}}{0.0091667}$$ Calculate: $$ (1+0.0091667)^{-36} = \frac{1}{(1.0091667)^{36}} \approx \frac{1}{1.432364} = 0.69855$$ $$PV = 20000 \times \frac{1-0.69855}{0.0091667} = 20000 \times \frac{0.30145}{0.0091667} \approx 20000 \times 32.88 = 657592.6$$ 4. **Analyze Payment Plan 3:** - Single payment R 900 000 at end of 3 years - Interest rate: 11% compounded monthly (monthly rate 0.0091667) Calculate the present value: $$PV = \frac{900000}{(1+0.0091667)^{36}} = \frac{900000}{1.432364} \approx 628384.8$$ 5. **Compare present values with debt:** - Plan 1 PV: R 400,526.32 (less than R 500,000) - Plan 2 PV: R 657,592.6 (more than R 500,000) - Plan 3 PV: R 628,384.8 (more than R 500,000) We want to pay closest to R 500,000 without paying more, so Plan 1 is closest but underpays. 6. **Adjust Plan 1 to exactly match R 500,000 debt:** Present value of first payment today is exact, so let the second payment amount be \(X\): $$100000 + \frac{X}{(1+0.10)^3} = 500000$$ $$\frac{X}{1.331} = 400000$$ $$X = 400000 \times 1.331 = 532400$$ So, to meet the debt exactly, you must pay R 100,000 today and R 532,400 after 3 years. **Final answer:** - Plan closest without overpaying: Plan 1 (present value ~R 400,526.32) - Amount to pay to meet debt exactly under Plan 1: R 100,000 today and R 532,400 at end of 3 years.