1. **Problem Statement:** Thomas owes R7000 in 3 years. He can pay R3000 in 1 year and a final payment $X$ in 30 months (2.5 years). The interest rate is 12.5% per year, compounded half-yearly. We need to find $X$ such that the present values of payments equal the present value of the debt. 2. **Formula and Important Rules:** - The interest rate per half-year period is $i = \frac{12.5\%}{2} = 6.25\% = 0.0625$. - Number of half-year periods for each time point: - 1 year = 2 half-years - 2.5 years = 5 half-years - 3 years = 6 half-years - Present value (PV) of a future amount $A$ at $n$ half-years is: $$PV = \frac{A}{(1+i)^n}$$ 3. **Calculate Present Value of Debt:** $$PV_{debt} = \frac{7000}{(1+0.0625)^6}$$ Calculate denominator: $$(1.0625)^6 \approx 1.4447$$ So, $$PV_{debt} = \frac{7000}{1.4447} \approx 4843.5$$ 4. **Calculate Present Value of First Payment:** $$PV_{3000} = \frac{3000}{(1.0625)^2}$$ Calculate denominator: $$(1.0625)^2 = 1.1289$$ So, $$PV_{3000} = \frac{3000}{1.1289} \approx 2656.3$$ 5. **Calculate Present Value of Final Payment $X$:** $$PV_X = \frac{X}{(1.0625)^5}$$ Calculate denominator: $$(1.0625)^5 \approx 1.3533$$ 6. **Set Present Values Equal:** $$PV_{debt} = PV_{3000} + PV_X$$ $$4843.5 = 2656.3 + \frac{X}{1.3533}$$ 7. **Solve for $X$:** $$\frac{X}{1.3533} = 4843.5 - 2656.3 = 2187.2$$ $$X = 2187.2 \times 1.3533 \approx 2958.5$$ 8. **Interpretation:** The final payment $X$ must be approximately R2958.5 to discharge the debt under the given conditions. **Note:** The option given (a) R4798.00 does not match this calculation, so the correct value is approximately R2959 when rounded to the nearest rand.