PRECAL HW question

Question

1. Home Mortgage A $120,000 home mortgage for 35 years at 712% has a monthly payment of $809.39. Part of the monthly payment is paid toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that is paid toward the interest is

U=(M-Pr/12)(1+r/12)^12t

and the amount that is paid toward the reduction of the principal is

v=(M-Pr/12)(1+r/12)^12t

In these formulas, P is the size of the mortgage, r is the interest rate, M is the monthly payment, and t is the time in years.

(a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.)

(b) In the early years of the mortgage, is the larger part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction.

(c) Repeat parts (a) and (b) for a repayment period of 20 years 􏰈M = $966.71􏰉. What can you conclude?

Answer 1

What's given in the problem 

1. Mortgage amount:
2. P=$120,000
3. Interest rate:
4. r=7.5%=0.075
5. Monthly payment (35 years):
6. M35=$809.39
7. Monthly payment (20 years):
8. M20=$966.71
9. Interest portion formula:
10. U=M−((M−Pr)/12) (1+r/12) ^12t
11. Principal portion formula:
12. V=((M−Pr)/12) (1+r/12) ^12t - Pr/12

U= 809.39-((809.39-120000*0.075)/12)(1+0.075/12)^12t

U= 809.39+-59.39(1.00625)^12t

V= (809.39-(120000*0.075)/12)(1+0.075/12)^12t

V= (59.39(1.00625)^12t )-750

Graph U and V from t=0 to t= 35

In early years U>V and the intersection point U=V is where they are equal.

so,

809.39-59.39*1.00625^12t = 59.39(1.00625)^12t -750

1559.39= 118.78(1.00625)^12t

1559.39/118.78 = 1.00625^12t

13.128=1.00625^12t

ln(13.1280= 12t ln(1.00625)

t= ln13.128/12 ln(1.00625)

t= approx 21.3 yrs

Repeat for 20 year graph where U=V

For the 35-year mortgage, interest is greater than principal in the early years, and they are approximately equal after

21.3 years.

For the 20-year mortgage, interest is greater than principal in the early years, and they are approximately equal after

18.6 years. Shorter loan terms result in the interest and principal being equal sooner.