1. **Problem statement:** You want to buy a home priced at $180000. You plan to pay 20% as a down payment and take a 30-year loan for the remaining amount. We need to find: a) The loan amount. b) The monthly payment if the interest rate is 6%. c) The monthly payment if the interest rate is 7%. 2. **Loan amount calculation:** The down payment is 20% of $180000, so the loan amount is the remaining 80%: $$\text{Loan amount} = 180000 \times (1 - 0.20) = 180000 \times 0.80 = 144000$$ 3. **Monthly payment formula:** The monthly payment for a fixed-rate mortgage is given by: $$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $M$ is the monthly payment - $P$ is the loan amount - $r$ is the monthly interest rate (annual rate divided by 12) - $n$ is the total number of payments (loan term in months) 4. **Calculate monthly payments for 6% interest:** - Annual interest rate = 6% = 0.06 - Monthly interest rate $r = \frac{0.06}{12} = 0.005$ - Number of payments $n = 30 \times 12 = 360$ Substitute values: $$M = 144000 \times \frac{0.005(1+0.005)^{360}}{(1+0.005)^{360} - 1}$$ Calculate $(1+0.005)^{360}$: $$1.005^{360} \approx 6.022575$$ So: $$M = 144000 \times \frac{0.005 \times 6.022575}{6.022575 - 1} = 144000 \times \frac{0.030112875}{5.022575}$$ Simplify fraction: $$\frac{0.030112875}{5.022575} \approx 0.005995$$ Therefore: $$M = 144000 \times 0.005995 = 863.28$$ 5. **Calculate monthly payments for 7% interest:** - Annual interest rate = 7% = 0.07 - Monthly interest rate $r = \frac{0.07}{12} \approx 0.0058333$ - Number of payments $n = 360$ Substitute values: $$M = 144000 \times \frac{0.0058333(1+0.0058333)^{360}}{(1+0.0058333)^{360} - 1}$$ Calculate $(1+0.0058333)^{360}$: $$1.0058333^{360} \approx 10.677$$ So: $$M = 144000 \times \frac{0.0058333 \times 10.677}{10.677 - 1} = 144000 \times \frac{0.0623}{9.677}$$ Simplify fraction: $$\frac{0.0623}{9.677} \approx 0.00644$$ Therefore: $$M = 144000 \times 0.00644 = 927.36$$ **Final answers:** - a) Loan amount = 144000 - b) Monthly payment at 6% interest = 863.28 - c) Monthly payment at 7% interest = 927.36