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.)

To determine the monthly payment for the 20-year mortgage option using the PMT formula, we first need to calculate the down payment and mortgage amount. The price of the cabin is \( \$45,000 \). The down payment at \( 5\% \) is: \[ \text{Down Payment} = 45000 \times 0.05 = 2250 \] The mortgage amount (loan principal) after the down payment is: \[ \text{Mortgage Amount} = 45000 - 2250 = 42750 \] For a 20-year fixed mortgage at \( 10\% \) (or \( 0.10 \)), with \( n = 12 \) (monthly payments), and \( t = 20 \): \[ PMT = \frac{42750 \left(\frac{0.10}{12}\right)}{1 - \left(1 + \frac{0.10}{12}\right)^{-12 \times 20}} \] Calculating the values: \[ PMT = \frac{42750 \cdot \left(\frac{0.10}{12}\right)}{1 - \left(1+\frac{0.10}{12}\right)^{-240}} \] First, calculate \( \frac{0.10}{12} = 0.0083333 \), then: \[ PMT = \frac{42750 \cdot 0.0083333}{1 - (1 + 0.0083333)^{-240}} \] Calculating \( (1 + 0.0083333)^{-240} \): \[ (1 + 0.0083333)^{-240} \approx 0.122042 \] Thus, \[ PMT \approx \frac{42750 \cdot 0.0083333}{1 - 0.122042} = \frac{356.25}{0.877958} \approx 405.25 \] Rounding to the nearest dollar, the monthly payment is: \[ PMT \approx \$405 \] To find the total interest paid over the mortgage period, calculate the total payments minus the loan amount: \[ \text{Total Payments} = PMT \times \text{total months} = 405 \times 240 = 97200 \] \[ \text{Total Interest} = \text{Total Payments} - \text{Loan Amount} = 97200 - 42750 = 54450 \] Now comparing it with the 30-year mortgage option: Using the same steps for a 30-year mortgage: \[ PMT = \frac{42750 \left(\frac{0.10}{12}\right)}{1 - \left(1 + \frac{0.10}{12}\right)^{-12 \times 30}} \] Calculate \( t = 30 \): Calculating \( (1 + 0.0083333)^{-360} \) gives approximately \( 0.0299614 \): \[ PMT \approx \frac{42750 \cdot 0.0083333}{1 - 0.0299614} \approx \frac{356.25}{0.9700386} \approx 367.17 \] Rounding gives \( PMT \approx \$367 \). Finding total interest for the 30-year option: \[ \text{Total Payments} = 367 \times 360 = 132120 \] \[ \text{Total Interest} = 132120 - 42750 = 89400 \] Finally, the savings in interest with the 20-year option compared to the 30-year option: \[ \text{Savings in Interest} = 89400 - 54450 = 34950 \] In conclusion: - The monthly payment for the 20-year option is \( \boxed{405} \). - The buyer saves \( \boxed{34950} \) in interest with the 20-year option.