The price of a home is \( \$ 200,000 \). 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. 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. \( \$ \square \) (Round to the nearest dollar as needed.)

To determine the down payment, we first calculate \( 15\% \) of the home price: Down Payment = \( 0.15 \times 200,000 = \$30,000 \). Thus, the mortgage amount (P) = \( 200,000 - 30,000 = \$170,000 \). Next, we can calculate the monthly payment (PMT) for both mortgage options using the given formula. For the 15-year mortgage: - \( r = 0.065 \) (annual interest rate) - \( n = 12 \) (number of months in a year) - \( t = 15 \) (loan term in years) Substituting these values into the formula gives us: PMT = \( \frac{170,000 \left(\frac{0.065}{12}\right)}{[1-\left(1+\frac{0.065}{12}\right)^{-12 \times 15}]} \). Calculating the monthly payment: 1. Convert \( \frac{0.065}{12} \) to a monthly rate which is approximately \( 0.0054167 \). 2. \( (1 + 0.0054167)^{-180} \) calculates to \( 0.329601 \). 3. Substitute the values back to solve for PMT: \[ PMT = \frac{170,000 \times 0.0054167}{1 - 0.329601} \approx \frac{920.671}{0.670399} \approx 1372.54 \] So, the monthly payment for the 15-year option is approximately \( \boxed{1373} \). ### Interest Calculations: Now, calculating total payments and interest for both options: **15-Year Mortgage:** - Total Payments = \( PMT \times 180 = 1373 \times 180 \approx \$247,140 \) - Interest Paid = Total Payments - Mortgage Amount = \( 247,140 - 170,000 \approx \$77,140 \) **30-Year Mortgage:** - For 30 years: \( t = 30 \), using a similar calculation: - PMT = \( \frac{170,000 \left(\frac{0.065}{12}\right)}{[1-\left(1+\frac{0.065}{12}\right)^{-12 \times 30}]} \). Calculating the monthly payment for the 30-year option gives approximately \( 1070 \). - Total Payments = \( 1070 \times 360 = 385,200 \) - Interest Paid = Total Payments - Mortgage Amount = \( 385,200 - 170,000 = 215,200 \) ### Interest Savings: - Savings in interest = Interest for 30-year - Interest for 15-year - Savings = \( 215,200 - 77,140 \approx 138,060 \) In summary, the monthly payment for the 15-year option is roughly \( \$1373 \), and the buyer saves about \( \$138,060 \) in interest with the quicker mortgage term!