1. **State the problem:** John owes 10,000 due in 4 years and 3,000 due in 5 years. He wants to reschedule payments as X now, 2X in 3 years, and 3X in 5 years. The interest rate is 10% per annum compounded monthly. Find the value of X so the payments' present value equals the original debt's present value. 2. **Identify parameters:** - Annual nominal interest rate: $i = 0.10$ - Monthly interest rate: $i_m = \frac{0.10}{12} = 0.0083333$ - Number of months for each time point: - Now: 0 months - 3 years later: $36$ months - 4 years later: $48$ months - 5 years later: $60$ months 3. **Calculate present value (PV) of original debts:** - PV of 10,000 due in 4 years: $$PV_1 = \frac{10000}{(1 + 0.0083333)^{48}}$$ - PV of 3,000 due in 5 years: $$PV_2 = \frac{3000}{(1 + 0.0083333)^{60}}$$ Calculate each: $$PV_1 = \frac{10000}{(1.0083333)^{48}} = \frac{10000}{1.488864} = 6716.51$$ $$PV_2 = \frac{3000}{(1.0083333)^{60}} = \frac{3000}{1.647009} = 1821.89$$ 4. **Calculate PV of rescheduled payments:** Payments P are: - $X$ now (0 months) - $2X$ in 3 years (36 months) - $3X$ in 5 years (60 months) PV of rescheduled payments: $$PV_{rescheduled} = X + \frac{2X}{(1.0083333)^{36}} + \frac{3X}{(1.0083333)^{60}}$$ Calculate powers: $$(1.0083333)^{36} = 1.395624$$ $$(1.0083333)^{60} = 1.647009$$ Therefore, $$PV_{rescheduled} = X + \frac{2X}{1.395624} + \frac{3X}{1.647009} = X + 1.4339X + 1.8219X = 4.2558X$$ 5. **Equate PV of debts and rescheduled payments:** $$6716.51 + 1821.89 = 4.2558X$$ $$8538.40 = 4.2558X$$ 6. **Solve for X:** $$X = \frac{8538.40}{4.2558} = 2006.42$$ **Final answer:** John’s payments should be approximately: - Now: $2006.42$ - In 3 years: $2 \times 2006.42 = 4012.84$ - In 5 years: $3 \times 2006.42 = 6019.26$