PRECAL HW REVISED

Question

1. Home Mortgage A $120,000 home mortgage for 35 years at 7 1/2% ( seven and a half) 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

1. Here is the graph showing the distribution of mortgage payments towards interest and principal over 35 years:
2. 
3. (a) The graph shows both the interest (U) and principal (v) components of the monthly payment over the course of 35 years.

Step 2

1. (b) In the early years of the mortgage, the larger part of the monthly payment is paid toward the interest. The point at which the monthly payment is evenly divided between interest and principal reduction can be approximated from the graph where the two lines intersect.

Step 3

1. C). Now, let's repeat the same for a repayment period of 20 years with a monthly payment of $966.71.
2. Here is the graph showing the distribution of mortgage payments towards interest and principal over 20 years:
3. 
4. Explanation:
5. (a) The graph shows both the interest (U) and principal (v) components of the monthly payment over the course of 20 years.
6. (b) Similar to the 35 -year mortgage, in the early years of the 20 -year mortgage, the larger part of the monthly payment is paid toward the interest. The point at which the monthly payment is evenly divided between interest and principal reduction can be approximated from the graph where the two lines intersect.
7. Comparing the two scenarios, we can conclude that with a shorter repayment period ( 20 years instead of 35 ), the portion of the monthly payment that goes towards the principal increases more quickly. This means that the borrower is building equity in the home faster. However, the monthly payment is higher in the 20 -year mortgage scenario.

Answers

For both the 35 -year and 20 -year mortgages, the larger part of the monthly payment is paid toward the interest in the early years. Over time, a greater portion of the monthly payment goes towards reducing the principal.

The point at which the monthly payment is evenly divided between interest and principal reduction can be approximated from the graphs where the two lines intersect. This point comes sooner in the 20-year mortgage compared to the 35-year mortgage.

With a shorter repayment period ( 20 years instead of 35 ), the portion of the monthly payment that goes towards the principal increases more quickly. This means that the borrower is building equity in the home faster. However, the monthly payment is higher in the 20-year mortgage scenario.

Therefore, while a shorter mortgage term results in higher monthly payments, it also allows the borrower to pay off the loan faster and pay less in total interest over the life of the loan.