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 \( \$ 85,000 \). The bank requires a \( 5 \% \) down payment. The buyer is offered two mortgage options: 20-year fixed at \( 8 \% \) or 30 -year fixed at \( 8 \% \). 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 calculate the monthly payment using the PMT formula, we first need to determine the loan amount after the down payment. The cabin price is $85,000, and the down payment is 5% of that: \[ \text{Down Payment} = 0.05 \times 85,000 = 4,250 \] Now, subtract the down payment from the cabin price to find the loan amount: \[ \text{Loan Amount} = 85,000 - 4,250 = 80,750 \] For the 20-year mortgage option with an 8% interest rate (0.08 yearly), the monthly interest rate \( r/n \) will be: \[ r = 0.08, \quad n = 12, \quad t = 20 \quad \Rightarrow \quad \frac{r}{n} = \frac{0.08}{12} = 0.0066667 \] Now, calculate \( n \cdot t \): \[ n \cdot t = 12 \cdot 20 = 240 \] Using the PMT formula: \[ PMT = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}} = \frac{80,750 \left( \frac{0.08}{12} \right)}{1 - \left( 1 + \frac{0.08}{12} \right)^{-240}} \] Calculating: \[ PMT \approx \frac{80,750 \cdot 0.0066667}{1 - (1+0.0066667)^{-240}} \] \[ \approx \frac{538.33}{1 - (1.0066667)^{-240}} \approx \frac{538.33}{1 - 0.145} \approx \frac{538.33}{0.855} \approx 629.90 \text{ (rounded to the nearest dollar) } \] The monthly payment for the 20-year option is approximately **$630**. Next, to find the total interest paid: 1. **Total payment over 20 years:** \[ \text{Total Payment} = PMT \cdot n \cdot t = 630 \cdot 240 = 151,200 \] 2. **Total interest paid:** \[ \text{Total Interest} = \text{Total Payment} - \text{Principal} = 151,200 - 80,750 = 70,450 \] For the 30-year mortgage option: - Using similar calculations for a 30-year mortgage at 8% interest rate \( t = 30 \): \[ n \cdot t = 12 \cdot 30 = 360 \] Monthly interest remains the same: \[ PMT_{30} = \frac{80,750 \cdot 0.0066667}{1 - (1+0.0066667)^{-360}} \approx \frac{538.33}{0.843} \approx 638.82 \approx 639 \text{ (rounded to the nearest dollar) } \] - **Total payment over 30 years:** \[ \text{Total Payment for 30 years} = 639 \cdot 360 = 230,040 \] - **Total interest paid for 30 years:** \[ \text{Total Interest}_{30} = 230,040 - 80,750 = 149,290 \] To find savings in interest: \[ \text{Savings} = \text{Total Interest}_{30} - \text{Total Interest}_{20} = 149,290 - 70,450 = 78,840 \] **Final Summary:** Monthly payment for the 20-year option: \( \$ 630 \) Total interest savings by choosing the 20-year option: \( \$ 78,840 \)