you are purchasing a new home and will need a mortgage for $215,000.00.\nyour mortgage options are…

Explanation:

Step1: Define mortgage payment formula

The formula for fixed monthly mortgage payment is: $$M = P \times \frac{\frac{r}{12}\left(1+\frac{r}{12}\right)^n}{\left(1+\frac{r}{12}\right)^n - 1}$$ Where:

• $M$ = monthly payment
• $P$ = principal loan amount ($215,000)
• $r$ = annual interest rate (decimal)
• $n$ = total number of payments (years × 12)

Step2: Calculate Option1 payment

30-year loan, 6.6% rate: $r=0.066$, $n=30\times12=360$ $$M_1 = 215000 \times \frac{\frac{0.066}{12}\left(1+\frac{0.066}{12}\right)^{360}}{\left(1+\frac{0.066}{12}\right)^{360} - 1}$$ $$M_1 = 215000 \times \frac{0.0055(1.0055)^{360}}{(1.0055)^{360} - 1} \approx 215000 \times \frac{0.0055\times7.21}{7.21 - 1} \approx $1357.20$$ Total paid: $1357.20\times360 \approx $488,592$

Step3: Calculate Option2 payment

20-year loan, 6% rate: $r=0.06$, $n=20\times12=240$ $$M_2 = 215000 \times \frac{\frac{0.06}{12}\left(1+\frac{0.06}{12}\right)^{240}}{\left(1+\frac{0.06}{12}\right)^{240} - 1}$$ $$M_2 = 215000 \times \frac{0.005(1.005)^{240}}{(1.005)^{240} - 1} \approx 215000 \times \frac{0.005\times3.31}{3.31 - 1} \approx $1564.26$$ Total paid: $1564.26\times240 \approx $375,422.40$

Step4: Calculate Option3 payment

15-year loan, 5.4% rate: $r=0.054$, $n=15\times12=180$ $$M_3 = 215000 \times \frac{\frac{0.054}{12}\left(1+\frac{0.054}{12}\right)^{180}}{\left(1+\frac{0.054}{12}\right)^{180} - 1}$$ $$M_3 = 215000 \times \frac{0.0045(1.0045)^{180}}{(1.0045)^{180} - 1} \approx 215000 \times \frac{0.0045\times2.22}{2.22 - 1} \approx $1754.63$$ Total paid: $1754.63\times180 \approx $315,833.40$

Step5: Compare total cost

Option3 has the lowest total amount paid over the life of the mortgage, despite higher monthly payments.

Answer:

Option 3