1. **Stating the problem:** You want to buy a $207,000 home with a 20% down payment, and take a 30-year loan at 3.25% annual interest for the remaining amount. We calculate the loan amount, monthly payments, total interest for 30 years, then the monthly payment and interest for 15 years, and finally the interest saved. 2. **Calculate down payment and loan amount:** Down payment = 20% of 207000 = $${207000 \times 0.20 = 41400}$$ Loan amount = $${207000 - 41400 = 165600}$$ 3. **Calculate monthly payment for 30-year loan:** The loan is amortized monthly, so: Annual interest rate = 3.25%, monthly rate $$r = \frac{3.25}{100 \times 12} = 0.0027083$$ Number of payments $$n = 30 \times 12 = 360$$ Monthly payment $$M$$ formula: $$M = P \frac{r(1+r)^n}{(1+r)^n - 1}$$ Substitute: $$M = 165600 \times \frac{0.0027083 (1+0.0027083)^{360}}{(1+0.0027083)^{360} - 1}$$ Calculate $$ (1+0.0027083)^{360} \approx 2.761$$ So: $$M = 165600 \times \frac{0.0027083 \times 2.761}{2.761 - 1} = 165600 \times \frac{0.007476}{1.761} = 165600 \times 0.004244 = 702.53$$ Monthly payment approximately $702.53. 4. **Calculate total interest paid over 30 years:** Total amount paid = monthly payment × number of payments $$702.53 \times 360 = 252,911.10$$ Interest paid = total paid - principal $$252,911.10 - 165,600 = 87,311.10$$ 5. **Calculate monthly payment for 15-year loan:** Number of payments $$n = 15 \times 12 = 180$$ Monthly payment formula again: $$M = 165600 \times \frac{0.0027083 (1+0.0027083)^{180}}{(1+0.0027083)^{180} - 1}$$ Calculate $$ (1+0.0027083)^{180} \approx 1.601$$ So: $$M = 165600 \times \frac{0.0027083 \times 1.601}{1.601 - 1} = 165600 \times \frac{0.004337}{0.601} = 165600 \times 0.007216 = 1195.70$$ Monthly payment approximately $1195.70. 6. **Calculate total interest paid over 15 years:** Total amount paid = $1195.70 × 180 = 215,226$ Interest = $215,226 - 165,600 = 49,626$ 7. **Calculate interest saved by paying 15 years instead of 30 years:** $$87,311.10 - 49,626 = 37,685.10$$ You save approximately $37,685.10 in interest. **Final answers:** a) Loan amount: $165,600 b) Monthly payment (30 years): $702.53 c) Total interest (30 years): $87,311.10 d) Monthly payment (15 years): $1,195.70 e) Interest saved: $37,685.10