Question

A man buys a house for $255,000. He pays $40,000 down and takes out a mortgage at 6.9% on the balance. Find his monthly payment and the total amount of interest he will pay if the length of the mortgage is (a) 15 years; (b) 20 years; (c) 25 years. (d) When will half the 20-year loan be paid off? (a) For the 15-year mortgage, the man will make monthly payments of $| (Do not round until the final answer. Then round to the nearest cent.)

Answer 1

Loan payment = 255000 - 40000 = 215000

i = 6.9% / 12 = 0.575% per month

a.

T = 15*12 = 180 months

Monthly payment = 215000 * (A/P,0.5775%,180)

= 215000 * 0.00575 * ((1 + 0.00575)^180)/((1 + 0.00575)^180-1)

= 215000 * 0.00575 * ((1.00575)^180)/((1.00575)^180-1)

= 1920.48 (Nearest Cent)

Total interest paid = 1920.48 * 180 - 215000 = 130686.40

b.

T = 20*12 = 240 months

Monthly payment = 215000 * (A/P,0.575%,240)

= 215000 * 0.00575 * ((1 + 0.00575)^240)/((1 + 0.00575)^240-1)

= 215000 * 0.00575 * ((1.00575)^240)/((1.00575)^240-1)

= 1654.01 (Nearest Cent)

Total interest paid = 1654.01 * 240 - 215000 = 181962.40

c.

T = 25*12 = 300 months

Monthly payment = 215000 * (A/P,0.575%,300)

= 215000 * 0.00575 * ((1 + 0.00575)^300)/((1 + 0.00575)^300-1)

= 215000 * 0.00575 * ((1.00575)^300)/((1.00575)^300-1)

= 1505.89 (Nearest Cent)

Total interest paid = 1505.89 * 300 - 215000 = 236767

d.

Let n months are left to pay when half balance is paid off

half balance = 215000 / 2 = 107500

As per given condition

1654.01 * (P/A,0.575%,n) = 107500

(P/A,0.575%,n) = 107500 / 1654.01 = 64.993561

((1 + 0.00575)^n-1)/(0.00575 *(1 + 0.00575)^n) = 64.993561

((1.00575)^n-1)/(0.00575 *(1.00575)^n) = 64.993561

(1.00575)^n - 1 = 64.993561 * (0.00575 *(1.00575)^n)

(1.00575)^n - 1 = 0.373712976 * (1.00575)^n

(1.00575)^n = 1 / (1 - 0.373712976) = 1.59671199

taking log both sides

n = log 1.59671199 / log 1.00575 = 81.6157 months

Months to pay half balance = 240 - 81.6157 = 158.38 months