since they filed bankruptcy in the past, a couple ends up paying a 12% fixed rate for a 30 - year mortgage…

Explanation:

Step1: Calculate monthly - rate and number of payments

The number of years $n = 30$, so the number of months $m=30\times12 = 360$. The 12% annual - rate monthly rate $r_1=\frac{0.12}{12}=0.01$, and the 8% annual - rate monthly rate $r_2=\frac{0.08}{12}=\frac{0.08}{12}\approx0.00667$. The loan amount $P = 140000$.

Step2: Use the mortgage - payment formula $M = P\times\frac{r(1 + r)^m}{(1 + r)^m-1}$

For the 12% rate: $M_1=140000\times\frac{0.01(1 + 0.01)^{360}}{(1 + 0.01)^{360}-1}$ Let $x=(1 + 0.01)^{360}=1.01^{360}\approx35.94964$. $M_1=140000\times\frac{0.01\times35.94964}{35.94964 - 1}=140000\times\frac{0.3594964}{34.94964}\approx140000\times0.0102861\approx1440.054$. For the 8% rate: $M_2=140000\times\frac{\frac{0.08}{12}(1+\frac{0.08}{12})^{360}}{(1+\frac{0.08}{12})^{360}-1}$ Let $y=(1+\frac{0.08}{12})^{360}=(1 + 0.00667)^{360}\approx10.9357$. $M_2=140000\times\frac{\frac{0.08}{12}\times10.9357}{10.9357 - 1}=140000\times\frac{\frac{0.874856}{12}}{9.9357}=140000\times\frac{0.0729047}{9.9357}\approx140000\times0.0073378\approx1029.292$.

Step3: Calculate the difference

The difference $\Delta M = M_1 - M_2=1440.054-1029.292 = 410.762\approx412.79$.

Answer:

B. $$412.79$