Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly

Question

a. How long will it take to repay the loan?
b. How much will be the final payment?
c. Determine how much interest she will pay for her loan.

Answer 1

i = .04/12 = .00333... (I store this in my calculator's memory for maximum accuracy)
n is number of months, so
200,000 = 3000 (1 - 1.003333...^-n)/.003333...
.2222.... = 1 - 1.00333...^-n
1.003333.^-n = .77777...
take log of both sides and use log rules
-n log1.003333... = log .7777..
-n = log .777.../log 1.003333.. = -75.519..
So you will need 75 monthly payments of $3000 and a partial payment at month 76.

b) balance at the end of 75 months
= 200000(1.0033333..)^75 - 3000(1.0033333...^75 - 1)/.0033333...
= 1555.81
interest on that amount for 1 months = .003333..(1555.81) = 5.186

final partial payment = 1555.81 + 5.186 = $1560.99

c) this is actually an invalid question, since you would be adding up sums of money that are not at the same time spot.
nevertheless, they probably want:
interest paid = total amount paid - 200,000
= 75(3000) + 1560.99 - 200000 = $26,560.99

check my arithmetic