Answer
See below
Work Step by Step
(a)
Option-A
Compute the monthly payment for mortgage as shown below
\[\begin{align}
& PMT=\frac{P\left( \frac{r}{n} \right)}{\left[ 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right]} \\
& =\frac{\$100,000\left(\frac{0.085}{12}\right)}{\left[1-{{\left(1+\frac{0.085}{12}\right)}^{-12\left(30\right)}}\right]}\\&=\frac{\$708.33}{\left[1-{{\left(1.0071\right)}^{-360}}\right]}\\&=\$768.59\\&\simeq\$769\end{align}\]
So,
\[\begin{align}
& \text{Total Amount }=PMT\times n\times t \\
& =\$769\times12\times30\\&=\$276,840\end{align}\]
Compute the amount of interest as follows:
\[\begin{align}
& \text{Interest}=\text{Total amount}-\text{Principal} \\
& =\$276,840-\$100,000\\&=\$176,840\end{align}\]
Option-B
\[P=\$95,700*,r=7.5percent,t=30\text{years,}n=12\]
*Cost of three points is $3,000, closing cost $1,300, total sum \[\$1,300+\$3,000=\$4,300\]
Then Principal becomes\[\$100,000-\$4300=\$95,700\].
Compute the monthly payment for mortgage as shown below
\[\begin{align}
& PMT=\frac{P\left( \frac{r}{n} \right)}{\left[ 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right]} \\
& =\frac{\$95700\left(\frac{0.075}{12}\right)}{\left[1-{{\left(1+\frac{0.075}{12}\right)}^{-12\left(30\right)}}\right]}\\&=\frac{598.125}{\left[1-{{\left(1.00625\right)}^{-360}}\right]}\\&=\frac{598.125}{\left[1-0.10614\right]}\end{align}\]
\[\begin{align}
& =\$669.14\\&\simeq\$670\end{align}\]
So,
Compute the total amount as follows:
\[\begin{align}
& \text{Total Amount }=PMT\times n\times t \\
& =\$670\times12\times30\\&=\$241,200\end{align}\]
Compute the total interest amount as follows:
\[\begin{align}
& \text{The Interest}=\text{Total amount}-\text{Principal} \\
& =\$241,200-\$95,700\\&=\$145,500\end{align}\]
The Difference between the interest from option-A and option-B is $31,340
