1. **State the problem:** John owes P10,000 due in 4 years and P3,000 due in 5 years. He reschedules to pay three sums: $X$ now, $2X$ in 3 years, and $3X$ in 5 years. The interest rate is 10% per annum compounded monthly. We need to find $X$ such that the present value of the new payments equals the present value of the original debts. 2. **Calculate the present value (PV) of the original debts:** - Interest rate per month $i = \frac{10\%}{12} = 0.0083333$ (approx) - Number of months for 4 years $n_1 = 4 \times 12 = 48$ - Number of months for 5 years $n_2 = 5 \times 12 = 60$ PV of P10,000 due in 4 years: $$PV_1 = \frac{10000}{(1 + 0.0083333)^{48}}$$ PV of P3,000 due in 5 years: $$PV_2 = \frac{3000}{(1 + 0.0083333)^{60}}$$ Total PV of original debts: $$PV_{total} = PV_1 + PV_2$$ 3. **Calculate the present value of the rescheduled payments:** - Payment $X$ now has PV = $X$ - Payment $2X$ in 3 years (36 months): $$PV_{2X} = \frac{2X}{(1 + 0.0083333)^{36}}$$ - Payment $3X$ in 5 years (60 months): $$PV_{3X} = \frac{3X}{(1 + 0.0083333)^{60}}$$ Total PV of rescheduled payments: $$PV_{rescheduled} = X + \frac{2X}{(1 + 0.0083333)^{36}} + \frac{3X}{(1 + 0.0083333)^{60}}$$ 4. **Set the total present values equal and solve for $X$:** $$PV_{total} = PV_{rescheduled}$$ $$\Rightarrow PV_1 + PV_2 = X + \frac{2X}{(1 + 0.0083333)^{36}} + \frac{3X}{(1 + 0.0083333)^{60}}$$ 5. **Calculate numerical values:** Calculate $(1 + 0.0083333)^{48} \approx 1.488863$ and $(1 + 0.0083333)^{60} \approx 1.647009$ and $(1 + 0.0083333)^{36} \approx 1.395618$ Then: $$PV_1 = \frac{10000}{1.488863} \approx 6717.57$$ $$PV_2 = \frac{3000}{1.647009} \approx 1821.99$$ $$PV_{total} = 6717.57 + 1821.99 = 8539.56$$ 6. **Calculate the coefficients for $X$:** $$PV_{rescheduled} = X + \frac{2X}{1.395618} + \frac{3X}{1.647009} = X + 1.4337X + 1.8219X = 4.2556X$$ 7. **Solve for $X$:** $$4.2556X = 8539.56$$ $$X = \frac{8539.56}{4.2556} \approx 2006.88$$ 8. **Final payments:** - Now: $X = 2006.88$ - In 3 years: $2X = 4013.76$ - In 5 years: $3X = 6020.64$ **Answer:** John’s payments are approximately P2006.88 now, P4013.76 in 3 years, and P6020.64 in 5 years.