1. **State the problem:** Paul wants to withdraw 3500 every 6 months for 5 years starting 10 years from now. The bank interest rate is 3.00% compounded monthly. We need to find how much she must invest now. 2. **Identify the formulas:** - The investment grows for 10 years with monthly compounding. - Withdrawals form an annuity due (withdrawals at beginning of each period) for 5 years every 6 months. 3. **Convert interest rate and periods:** - Annual nominal rate $i = 0.03$ compounded monthly means monthly rate $i_m = \frac{0.03}{12} = 0.0025$. - Number of months in 10 years: $10 \times 12 = 120$. - Withdrawals every 6 months means 2 withdrawals per year, total withdrawals $5 \times 2 = 10$. - Effective 6-month interest rate from monthly compounding: $$i_{6m} = (1 + i_m)^6 - 1 = (1 + 0.0025)^6 - 1 = 1.015114 - 1 = 0.015114$$ 4. **Calculate the present value of the annuity at the time withdrawals start (10 years):** Annuity due present value formula: $$PV_{10} = R \times \frac{1 - (1 + i_{6m})^{-n}}{i_{6m}} \times (1 + i_{6m})$$ where $R=3500$, $n=10$, $i_{6m}=0.015114$. Calculate: $$1 - (1 + 0.015114)^{-10} = 1 - (1.015114)^{-10} = 1 - 0.860707 = 0.139293$$ So: $$PV_{10} = 3500 \times \frac{0.139293}{0.015114} \times 1.015114 = 3500 \times 9.217 \times 1.015114 = 3500 \times 9.356 = 32746.3$$ 5. **Calculate the amount to invest now (time 0) to have $PV_{10}$ in 10 years:** Using monthly compounding: $$PV_0 = \frac{PV_{10}}{(1 + i_m)^{120}} = \frac{32746.3}{(1.0025)^{120}}$$ Calculate denominator: $$(1.0025)^{120} = e^{120 \times \ln(1.0025)} \approx e^{120 \times 0.002496} = e^{0.2995} = 1.3499$$ So: $$PV_0 = \frac{32746.3}{1.3499} = 24250.3$$ 6. **Final answer:** Paul must invest approximately **24250.30** now to enable the withdrawals.