1. **Problem Statement:** Find the present deposit amount that will provide beginning-of-month payments of 600 for the first 2 years and 800 for the next 4 years, with an account earning 4% interest compounded quarterly. 2. **Given Data:** - Payments: 600 for 2 years, then 800 for 4 years - Interest rate: 4% compounded quarterly - Total duration: 6 years - Payments are at the beginning of each month 3. **Key Formulas and Concepts:** - Convert annual nominal interest rate compounded quarterly to effective monthly rate: $$ i = \left(1 + \frac{0.04}{4}\right)^{\frac{4}{12}} - 1 = \left(1 + 0.01\right)^{\frac{1}{3}} - 1 $$ - Number of payments: - First period: $2 \times 12 = 24$ payments - Second period: $4 \times 12 = 48$ payments - Present value of an annuity due (payments at beginning of period): $$ PV = PMT \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i) $$ 4. **Calculate effective monthly interest rate:** $$ i = (1.01)^{\frac{1}{3}} - 1 \approx 0.003322 $$ 5. **Calculate present value of first 2 years payments (600 each):** $$ PV_1 = 600 \times \frac{1 - (1 + 0.003322)^{-24}}{0.003322} \times (1 + 0.003322) $$ Calculate inside the fraction: $$ 1 - (1.003322)^{-24} = 1 - \frac{1}{(1.003322)^{24}} $$ Calculate $(1.003322)^{24} \approx 1.0834$, so: $$ 1 - \frac{1}{1.0834} = 1 - 0.9229 = 0.0771 $$ Then: $$ \frac{0.0771}{0.003322} \approx 23.22 $$ Multiply by $1.003322$: $$ 23.22 \times 1.003322 \approx 23.30 $$ Finally: $$ PV_1 = 600 \times 23.30 = 13980 $$ 6. **Calculate present value of next 4 years payments (800 each):** First find present value at the start of year 3 (after 24 months): $$ PV_2 = 800 \times \frac{1 - (1 + 0.003322)^{-48}}{0.003322} \times (1 + 0.003322) $$ Calculate inside the fraction: $$ (1.003322)^{48} \approx (1.0834)^2 = 1.173 $$ $$ 1 - \frac{1}{1.173} = 1 - 0.8529 = 0.1471 $$ Then: $$ \frac{0.1471}{0.003322} \approx 44.29 $$ Multiply by $1.003322$: $$ 44.29 \times 1.003322 \approx 44.44 $$ So: $$ PV_2 = 800 \times 44.44 = 35552 $$ 7. **Discount $PV_2$ back 24 months to present value:** $$ PV_2^{present} = \frac{35552}{(1.003322)^{24}} = \frac{35552}{1.0834} \approx 32825 $$ 8. **Total present value (deposit needed now):** $$ PV = PV_1 + PV_2^{present} = 13980 + 32825 = 46805 $$ 9. **Calculate total interest earned over 6 years:** Total payments made: $$ 600 \times 24 + 800 \times 48 = 14400 + 38400 = 52800 $$ Interest earned: $$ 52800 - 46805 = 5995 $$ **Final answers:** - Deposit needed now: $46805$ - Total interest earned: $5995$