Points: 0 of 1 Save \[ P\left(\frac{r}{n}\right) \] Use $\frac{r}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$ to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $\$ 195,000$ mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage A: 30 -year fixed at $9.25 \%$ with closing costs of $\$ 2700$ and 1 point. Mortgage B: 30 -year fixed at $7.5 \%$ with closing costs of $\$ 2700$ and 2 points. Choose the correct answer below, and fill in the answer box to complete your choice. (Do not round until the final answer. Then round to the nearest dollar as needed.) A. Mortgage $\mathrm{B}$ has a larger total cost than mortgage $\mathrm{A}$ by $\$$ B. Mortgage A has a larger total cost than mortgage $B$ by $\$$.

Step 1 ：First, let's calculate the regular payment amount for Mortgage A. The interest rate is 9.25%, which is 0.0925 in decimal form. The mortgage is for 30 years and there is one payment per year, so n=1 and t=30. The mortgage amount is $195,000. Using the formula, we get \(P_A = 195000 \times \left(\frac{0.0925}{1 - (1 + \frac{0.0925}{1})^{-1 \times 30}}\right) = 195000 \times \left(\frac{0.0925}{1 - 0.6068}\right) = 195000 \times \left(\frac{0.0925}{0.3932}\right) = 195000 \times 0.2353 = \$45,879.50\)

Step 2 ：The total cost of Mortgage A is the sum of the closing costs, the amount paid for points, and the total cost of interest. The amount paid for points is 1% of the mortgage amount, so it's $1950. So, the total cost of Mortgage A is \(\$2700 + \$1950 + \$45,879.50 \times 30 = \$1,395,135\)

Step 3 ：Next, let's calculate the regular payment amount for Mortgage B. The interest rate is 7.5%, which is 0.075 in decimal form. The mortgage is for 30 years and there is one payment per year, so n=1 and t=30. The mortgage amount is $195,000. Using the formula, we get \(P_B = 195000 \times \left(\frac{0.075}{1 - (1 + \frac{0.075}{1})^{-1 \times 30}}\right) = 195000 \times \left(\frac{0.075}{1 - 0.6966}\right) = 195000 \times \left(\frac{0.075}{0.3034}\right) = 195000 \times 0.2472 = \$48,204\)

Step 4 ：The total cost of Mortgage B is the sum of the closing costs, the amount paid for points, and the total cost of interest. The amount paid for points is 2% of the mortgage amount, so it's $3900. So, the total cost of Mortgage B is \(\$2700 + \$3900 + \$48,204 \times 30 = \$1,452,320\)

Step 5 ：Comparing the total costs, Mortgage B has a larger total cost than Mortgage A by \(\$1,452,320 - \$1,395,135 = \$57,185\)

Step 6 ：\(\boxed{\text{B. Mortgage A has a larger total cost than mortgage B by \$57,185.}}\)