Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest dollar. The price of a small cabin is \( \$ 45,000 \). The bank requires a \( 5 \% \) down payment. The buyer is offered two mortgage options: 20-year fixed at \( 10 \% \) or 30 -year fixed at 10\%. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option? Find the monthly payment for the 20-year option. \( \$ \square \) (Round to the nearest dollar as needed.)

Alright, I need to determine the regular payment amount for a small cabin priced at \$45,000 with a 5% down payment. The buyer has two mortgage options: a 20-year fixed at 10% and a 30-year fixed at 10%. I need to calculate the amount of interest paid for each option and find out how much the buyer saves in interest with the 20-year option. Additionally, I need to find the monthly payment for the 20-year option, rounded to the nearest dollar. First, let's break down the problem step by step. 1. **Calculate the Down Payment:** - The cabin costs \$45,000. - The down payment is 5% of \$45,000. - Down payment = 0.05 * 45,000 = \$2,250. 2. **Determine the Loan Amount:** - Loan amount = Total price - Down payment = 45,000 - 2,250 = \$42,750. 3. **Understand the Mortgage Options:** - **Option 1:** 20-year fixed at 10%. - **Option 2:** 30-year fixed at 10%. 4. **Calculate Monthly Interest Rate:** - Annual interest rate = 10%. - Monthly interest rate = 10% / 12 = 0.8333% or 0.008333 in decimal. 5. **Calculate the Number of Payments:** - **Option 1:** 20 years * 12 months/year = 240 payments. - **Option 2:** 30 years * 12 months/year = 360 payments. 6. **Use the PMT Formula:** - PMT = [P * (r/n)] / [1 - (1 + r/n)^(-n*t)] - Where: - P = Loan amount (\$42,750) - r = Annual interest rate (0.10) - n = Number of payments per year (12) - t = Total number of years (20 or 30) 7. **Calculate PMT for 20-Year Option:** - PMT = [42,750 * (0.10/12)] / [1 - (1 + 0.10/12)^(-12*20)] - PMT = [42,750 * 0.008333] / [1 - (1.008333)^(-240)] - First, calculate (1.008333)^(-240): - Using a calculator, (1.008333)^(-240) ≈ 0.2145 - Then, 1 - 0.2145 = 0.7855 - Now, PMT = (42,750 * 0.008333) / 0.7855 - 42,750 * 0.008333 ≈ 354.17 - 354.17 / 0.7855 ≈ 450.00 - So, the monthly payment for the 20-year option is approximately \$450. 8. **Calculate Total Interest Paid for 20-Year Option:** - Total payments = 240 * \$450 = \$108,000 - Total interest = Total payments - Loan amount = 108,000 - 42,750 = \$65,250 9. **Calculate PMT for 30-Year Option:** - PMT = [42,750 * (0.10/12)] / [1 - (1 + 0.10/12)^(-12*30)] - PMT = [42,750 * 0.008333] / [1 - (1.008333)^(-360)] - First, calculate (1.008333)^(-360): - Using a calculator, (1.008333)^(-360) ≈ 0.1667 - Then, 1 - 0.1667 = 0.8333 - Now, PMT = (42,750 * 0.008333) / 0.8333 - 42,750 * 0.008333 ≈ 354.17 - 354.17 / 0.8333 ≈ 425.00 - So, the monthly payment for the 30-year option is approximately \$425. 10. **Calculate Total Interest Paid for 30-Year Option:** - Total payments = 360 * \$425 = \$153,000 - Total interest = Total payments - Loan amount = 153,000 - 42,750 = \$110,250 11. **Determine Savings in Interest with 20-Year Option:** - Interest saved = Interest for 30-year option - Interest for 20-year option - Interest saved = 110,250 - 65,250 = \$45,000 12. **Finalize the Monthly Payment for the 20-Year Option:** - Rounded to the nearest dollar, the monthly payment is \$450.