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. \( \$ 1,481 \) (Round to the nearest dollar as needed.) Find the monthly payment for the 30 -year option. \( \$ 1,075 \) (Round to the nearest dollar as needed.) Calculate the total cost of interest for both mortgage options. How much does the buyer save in interest with the 15 -year option? \( \$ \square \) (Use the answers from parts 1 and 2 to find this answer.)

The home’s price is \( \$200,\!000 \) with a \( 15\% \) down payment. First, we determine the mortgage principal: \[ \text{Down Payment} = 0.15 \times \$200,\!000 = \$30,\!000 \] \[ \text{Loan Amount} = \$200,\!000 - \$30,\!000 = \$170,\!000 \] The monthly payment formulas have already been applied, yielding: - 15-year option: \( \text{PMT}_{15} = \$1,\!481 \) - 30-year option: \( \text{PMT}_{30} = \$1,\!075 \) ### 1. Total Interest for the 15-Year Option Number of months in 15 years: \[ n_{15} = 15 \times 12 = 180 \] Total amount paid over 15 years: \[ \text{Total Paid}_{15} = \$1,\!481 \times 180 = \$266,\!580 \] Interest paid: \[ \text{Interest}_{15} = \$266,\!580 - \$170,\!000 = \$96,\!580 \] ### 2. Total Interest for the 30-Year Option Number of months in 30 years: \[ n_{30} = 30 \times 12 = 360 \] Total amount paid over 30 years: \[ \text{Total Paid}_{30} = \$1,\!075 \times 360 = \$387,\!000 \] Interest paid: \[ \text{Interest}_{30} = \$387,\!000 - \$170,\!000 = \$217,\!000 \] ### 3. Interest Savings with the 15-Year Option The savings in interest by choosing the 15-year mortgage over the 30-year mortgage is: \[ \text{Savings} = \text{Interest}_{30} - \text{Interest}_{15} = \$217,\!000 - \$96,\!580 = \$120,\!420 \] Thus, the buyer saves \(\$120,\!420\) in interest with the 15-year option.