Sure, let's compute the true annual interest rate step-by-step using the provided formula:
[tex]\[
I = \frac{2yc}{m(n+1)}
\][/tex]
Given values:
- Loan amount ([tex]\(y\)[/tex]): \[tex]$8,500
- Monthly payments (\(c\)): \$[/tex]170.50
- Time of loan contract ([tex]\(n\)[/tex]): 5 years
First, let's determine the total number of monthly installments over the loan period. Since there are 12 months in a year:
[tex]\[
m = n \times 12 = 5 \times 12 = 60 \text{ months}
\][/tex]
We substitute the known values into the formula:
[tex]\[
I = \frac{2 \times 8500 \times 170.50}{60 \times (5 + 1)}
\][/tex]
Simplify inside the denominator first:
[tex]\[
I = \frac{2 \times 8500 \times 170.50}{60 \times 6}
\][/tex]
Now calculate [tex]\(60 \times 6\)[/tex]:
[tex]\[
60 \times 6 = 360
\][/tex]
The formula now becomes:
[tex]\[
I = \frac{2 \times 8500 \times 170.50}{360}
\][/tex]
Compute the numerator ([tex]\(2 \times 8500 \times 170.50\)[/tex]):
[tex]\[
2 \times 8500 = 17000
\][/tex]
[tex]\[
17000 \times 170.50 = 2,898,500
\][/tex]
Now, divide the result by 360:
[tex]\[
I = \frac{2898500}{360} = 8051.39
\][/tex]
So, the true annual interest rate is approximately [tex]\(8051.39\)[/tex].
*If you were asked to express this as a percentage (which actually represents usual financial interest rate), it would typically be converted to a rate per year and expressed as a small percentage. However, if not, the calculations are complete here, ensuring the total amount repaid over the period as such.
