Metin E. answered • 03/29/24
MS in Statistics, taught Finite Math for 2 years at community college
I will first rewrite the question in a clearer way.
"Find the amount applied to the principal for the third month of a 4-year loan of $22,300
which charges 5.7% compounded monthly with monthly payments."
We must start by determining the fixed monthly payments necessary to repay this loan in 4 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 / 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 = $22,300
r = 5.7% = 0.057
m = 12 because there will be monthly payments and there are 12 months in a year
t = 4 because the loan is to repaid in 4 years.
When we put all of this together, we get:
R = 22,300 * (0.057 / 12) / [1 - (1 + 0.057 / 12)^(-12 * 4)]
We can take a moment to find the monthly interest rate
0.057 / 12 = 0.00475
and the number of months in 4 years
12 * 4 = 48
to make the formula a little simpler
R = 22,300 * 0.00475 / [1 - (1 + 0.00475)^(-48)]
We now just put this into the calculator and get:
R ≈ $520.65
We can now make an amortization table:
Month Beginning Balance Interest Charged "Middle" Balance Ending Balance
1 $22,300.00 $105.93 $22,405.93 $21,885.28
2 $21,885.28 $103.96 $21,989.24 $21,468.59
3 $21,468.59 $101.98 $21,570.57 $21,049.92
How does the table work?
- The interest charged is found by multiplying the beginning balance by 0.057 / 12 or 0.00475.
- The middle balance is the sum of the beginning balance and the interest charged.
- The ending balance is the middle balance minus a monthly payment of $520.65.
- The beginning balance of the next month is the same as the ending balance of the previous month.
How do we find the amount applied to the principal in the 3rd month?
There are 2 methods.
Method 1:
The interest charged during the 3rd month was $101.98.
So out of the $520.65 payment, $101.98 would go towards the interest and the rest towards the principal.
The amount going towards the principal is:
$520.65 - $101.98 = $418.67
Method 2:
The beginning balance of month 3 was $21,468.59;
the ending balance is $21,049.92.
The difference was taken care of by the monthly payment.
That is, the amount of the monthly payment that went towards the principal is:
$21,468.59 - $21,049.92 = $418.67
