1. **Problem Statement:** Isaac wants to purchase a 25-year annuity with monthly payments of 1000 for the first 15 years and 1500 for the remaining 10 years. The interest rate is 4.8% compounded monthly. We need to find the present value (price) of this annuity. 2. **Formula and Concepts:** The present value of an annuity with monthly payments is given by: $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ where $P$ is the monthly payment, $i$ is the monthly interest rate, and $n$ is the total number of payments. 3. **Calculate monthly interest rate:** $$i = \frac{4.8\%}{12} = \frac{0.048}{12} = 0.004$$ 4. **Calculate number of payments:** - For first 15 years: $n_1 = 15 \times 12 = 180$ - For next 10 years: $n_2 = 10 \times 12 = 120$ 5. **Calculate present value of first 15 years payments:** $$PV_1 = 1000 \times \frac{1 - (1 + 0.004)^{-180}}{0.004}$$ Calculate $(1 + 0.004)^{-180} = (1.004)^{-180}$. 6. **Calculate present value of next 10 years payments:** Payments start after 15 years, so discount back 15 years (180 months) additionally. First, calculate present value of 1500 payments for 120 months: $$PV_2 = 1500 \times \frac{1 - (1 + 0.004)^{-120}}{0.004}$$ Then discount $PV_2$ back 180 months: $$PV_2 = PV_2 \times (1 + 0.004)^{-180}$$ 7. **Calculate values:** - $(1.004)^{-180} = \frac{1}{(1.004)^{180}} \approx \frac{1}{2.030856} = 0.4927$ - $(1.004)^{-120} = \frac{1}{(1.004)^{120}} \approx \frac{1}{1.601032} = 0.6247$ Calculate $PV_1$: $$PV_1 = 1000 \times \frac{1 - 0.4927}{0.004} = 1000 \times \frac{0.5073}{0.004} = 1000 \times 126.825 = 126825$$ Calculate $PV_2$ before discounting: $$PV_2 = 1500 \times \frac{1 - 0.6247}{0.004} = 1500 \times \frac{0.3753}{0.004} = 1500 \times 93.825 = 140737.5$$ Discount $PV_2$ back 180 months: $$PV_2 = 140737.5 \times 0.4927 = 69344.5$$ 8. **Total present value (price) of annuity:** $$PV = PV_1 + PV_2 = 126825 + 69344.5 = 196169.5$$ **Final answer:** Isaac will pay approximately 196170 for the annuity.