Metin E. answered • 03/27/24
MS in Statistics, taught Finite Math for 2 years at community college
Let us look at one thing at a time.
"A student borrows $52,000 at 7.2% interest. Find the monthly payment to repay the loan over 15 years".
This is a classical amortization of a loan and there is a formula to find fixed monthly payments in this context.
The formula is:
R = P(r / m) / [1 - (1 + r / m)^(-mt)]
where
P = the original loan amount (the letter P is used because this amount was the principal)
r = nominal interest rate per year
m = number of payments /conversion periods per year
t = the term of the loan (number of years)
R = amount of each payment
So in this problem:
P = $52,000 because the student borrows that amount
r = 7.2% = 0.072
m = 12 because there will be monthly payments and there are 12 months in a year
t = 15 because the loan is to repaid in 15 years.
When we put all of this together, we get:
R = 52,000 * (0.072 / 12) / [1 - (1 + 0.072 / 12)^(-12 * 15)]
We can take a moment to find the monthly interest rate
0.072 / 12 = 0.006
and the number of months in 15 years
12 * 15 = 180
to make the formula a little simpler
R = 52,000 * 0.006 / [1 - (1 + 0.006)^(-180)]
We now just put this into the calculator and get:
R ≈ $473.22
The student should make monthly payments of $473.22 for 15 years to repay the loan.
On to the second part...
"Find the total interest paid over the 15 year payment plan."
In doing monthly payments of $473.22 for 15 years, how much does the student pay in total?
$473.22 for 12 months for 15 years... so
$473.22 * 12 * 15 = $85,179.60
The student borrowed $52,000 but ended up repaying $85,179.60 !!!
Where does the huge difference come from?
That is the money that went to interest.
So the total interest paid over the 15 year payment plan is:
$85,179.60 - $52,000 = $33,179.60
