1. **Problem statement:** Sadie wants to save enough money to receive monthly payments of 3000 for 40 years. The interest rate is 3% per year compounded monthly. We need to find (a) the amount she must save initially and (b) the total interest earned. 2. **Formula used:** This is a present value of an annuity problem. The formula for the present value $PV$ of an annuity paying $P$ per period for $n$ periods at interest rate $i$ per period is: $$PV = P \times \frac{1 - (1+i)^{-n}}{i}$$ where: - $P = 3000$ (monthly payment) - $i = \frac{0.03}{12} = 0.0025$ (monthly interest rate) - $n = 40 \times 12 = 480$ (total number of monthly payments) 3. **Calculate present value:** $$PV = 3000 \times \frac{1 - (1+0.0025)^{-480}}{0.0025}$$ Calculate $(1+0.0025)^{-480}$: $$1.0025^{-480} = \frac{1}{1.0025^{480}}$$ Calculate $1.0025^{480}$: $$1.0025^{480} = e^{480 \times \ln(1.0025)} \approx e^{480 \times 0.0024969} = e^{1.1985} \approx 3.315$$ So: $$1.0025^{-480} = \frac{1}{3.315} \approx 0.3017$$ Now substitute back: $$PV = 3000 \times \frac{1 - 0.3017}{0.0025} = 3000 \times \frac{0.6983}{0.0025}$$ Simplify fraction: $$\frac{0.6983}{0.0025} = 279.32$$ So: $$PV = 3000 \times 279.32 = 837,960$$ 4. **Interpretation:** Sadie needs to save approximately 837,960 initially to receive 3000 monthly for 40 years at 3% annual interest compounded monthly. 5. **Calculate total amount paid out:** Total payments = $3000 \times 480 = 1,440,000$ 6. **Calculate interest earned:** Interest = Total payments - Present value $$= 1,440,000 - 837,960 = 602,040$$ **Final answers:** (a) Sadie must save approximately **837,960** initially. (b) She earns approximately **602,040** in interest over 40 years.