1. **State the problem:** You want to buy a home for $220000, with a 10% down payment. You take a loan for the remainder at 3.85% annual interest, compounded monthly. The loan term is 30 years, and you want to find the loan amount, monthly payments, total interest paid, monthly payments if paid off in 15 years, and interest saved. 2. **Calculate the loan amount (a):** Down payment = 10% of $220000 = $220000 \times 0.10 = $22000 Loan amount = Home price - Down payment = $220000 - $22000 = $198000 3. **Monthly payment for 30-year loan (b):** Loan principal $P = 198000$ Annual interest rate $r = 3.85\% = 0.0385$ Monthly interest rate $i = \frac{r}{12} = \frac{0.0385}{12} \approx 0.0032083$ Number of monthly payments for 30 years $n = 30 \times 12 = 360$ Use the amortization formula for monthly payment $M$: $$ M = P \times \frac{i(1+i)^n}{(1+i)^n - 1} $$ Calculate: $$ M = 198000 \times \frac{0.0032083 (1+0.0032083)^{360}}{(1+0.0032083)^{360} - 1} $$ Estimate $ (1+0.0032083)^{360} \approx e^{0.0032083 \times 360} = e^{1.155} \approx 3.174$ Thus, $$ M = 198000 \times \frac{0.0032083 \times 3.174}{3.174 - 1} = 198000 \times \frac{0.01018}{2.174} \approx 198000 \times 0.00468 = 926.64 $$ So, monthly payment $\approx 927$ 4. **Total interest paid for 30 years (c):** Total paid = Monthly payment $\times$ number of payments = $926.64 \times 360 = 333,590.4$ Interest paid = Total paid - Principal = $333,590.4 - 198000 = 135,590.4$ 5. **Monthly payment for 15-year loan (d):** Number of monthly payments $n = 15 \times 12 = 180$ Recalculate $M$ with $n=180$: $$ M = 198000 \times \frac{0.0032083 (1+0.0032083)^{180}}{(1+0.0032083)^{180} -1} $$ Estimate $ (1+0.0032083)^{180} \approx e^{0.0032083 \times 180} = e^{0.5775} \approx 1.781$ Then, $$ M = 198000 \times \frac{0.0032083 \times 1.781}{1.781 -1} = 198000 \times \frac{0.00571}{0.781} \approx 198000 \times 0.00731 = 1446.38 $$ Monthly payment $\approx 1446$ 6. **Interest saved by paying over 15 years instead of 30 years (e):** Total paid over 15 years = $1446.38 \times 180 = 260,348.4$ Interest paid over 15 years = $260,348.4 - 198000 = 62,348.4$ Interest saved = Interest over 30 years - Interest over 15 years $$ 135,590.4 - 62,348.4 = 73,242 $$ **Final answers:** (a) Loan amount = $198000$ (b) Monthly payment (30 years) $\approx 927$ (c) Total interest (30 years) $\approx 135590$ (d) Monthly payment (15 years) $\approx 1446$ (e) Interest saved by 15-year payoff $\approx 73242$