The price of a home is \( \$ 230,000 \). The bank requires a \( 15 \% \) down payment. The buyer is offered two mortgage options: 15 -year fixed at \( 6.5 \% \) or 30 -year fixed at \( 6.5 \% \). Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15 -year option? Use the following formula to determine the regular payment amount. \[ \text { PMT }=\frac{\mathrm{P}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \] Find the monthly payment for the 15-year option. \( \$ 1,703 \) (Round to the nearest dollar as needed.) Find the monthly payment for the 30-year option. \( \$ \square \) (Round to the nearest dollar as needed.)

To find the monthly payment for the 30-year option, we'll first determine the loan amount after the 15% down payment. The down payment amount is \(0.15 \times 230,000 = \$34,500\). Therefore, the loan amount is \(230,000 - 34,500 = \$195,500\). Now, for the 30-year mortgage at \(6.5\%\), let's find the values needed for our formula: - \(P = 195,500\) - \(r = 0.065\) (annual interest rate) - \(n = 12\) (number of payments per year) - \(t = 30\) (years) Now plug those values into the formula: \[ \text{PMT} = \frac{195,500 \left(\frac{0.065}{12}\right)}{1 - \left(1 + \frac{0.065}{12}\right)^{-12 \times 30}} \] Calculating this: \[ \text{PMT} = \frac{195,500 \times 0.00541667}{1 - (1 + 0.00541667)^{-360}} \] Calculating the denominator, \[ 1 - (1.00541667)^{-360} \approx 0.835279 \] Now calculate the payment: \[ \text{PMT} \approx \frac{195,500 \times 0.00541667}{0.835279} \approx \$1,009 \quad \text{(rounded to the nearest dollar)} \] Now that we know the monthly payment for the 30-year option is approximately \( \$1,009 \). To find out how much interest is paid for each option, we need to calculate the total payment made over the term of the loans and subtract the principal. 1. **For the 15-year mortgage:** - Total payment: \( \$1,703 \times 15 \times 12 = \$306,540\) - Total interest paid: \( \$306,540 - \$195,500 = \$111,040\) 2. **For the 30-year mortgage:** - Total payment: \( \$1,009 \times 30 \times 12 = \$362,040\) - Total interest paid: \( \$362,040 - \$195,500 = \$166,540\) **Savings in interest with the 15-year option:** \[ \$166,540 - \$111,040 = \$55,500 \] **To summarize:** - Monthly payment for the 30-year option: \( \$1,009 \) - Total interest saved by choosing the 15-year option: \( \$55,500 \)