the smiths were just approved for a 25 - year mortgage at an 11% fixed rate. if they had not filed…

Explanation:

Step1: Recall mortgage - payment formula

The formula for the monthly payment of a mortgage is $M = P\frac{r(1 + r)^n}{(1 + r)^n-1}$, where $M$ is the monthly payment, $P$ is the principal loan amount, $r$ is the monthly interest rate, and $n$ is the total number of payments.

Step2: Calculate the monthly interest rate and number of payments

The loan term is 25 years. So the number of payments $n=25\times12 = 300$ months. For the 11% rate: The annual interest rate $i_1 = 0.11$, so the monthly interest rate $r_1=\frac{0.11}{12}$. For the 7% rate: The annual interest rate $i_2 = 0.07$, so the monthly interest rate $r_2=\frac{0.07}{12}$. The principal amount $P = 128000$.

Step3: Calculate the monthly payment at 11%

$M_1=128000\times\frac{\frac{0.11}{12}(1 + \frac{0.11}{12})^{300}}{(1+\frac{0.11}{12})^{300}-1}$ Let $x=\frac{0.11}{12}\approx0.009167$. Then $(1 + x)^{300}=(1 + 0.009167)^{300}\approx13.2677$. $M_1=128000\times\frac{0.009167\times13.2677}{13.2677 - 1}=128000\times\frac{0.1216}{12.2677}\approx1265.57$.

Step4: Calculate the monthly payment at 7%

$M_2=128000\times\frac{\frac{0.07}{12}(1+\frac{0.07}{12})^{300}}{(1+\frac{0.07}{12})^{300}-1}$ Let $y = \frac{0.07}{12}\approx0.005833$. Then $(1 + y)^{300}=(1+0.005833)^{300}\approx5.4274$. $M_2=128000\times\frac{0.005833\times5.4274}{5.4274 - 1}=128000\times\frac{0.03165}{4.4274}\approx915.71$.

Step5: Calculate the difference in monthly payments

The difference $\Delta M=M_1 - M_2=1265.57-915.71 = 349.86$.

Answer:

A. $349.86$