To solve this problem, we need to compute the interest using the given formula:
[tex]\[
I = \frac{2 \times y \times c}{m \times (n + 1)}
\][/tex]
where:
- [tex]\( I \)[/tex] is the interest
- [tex]\( y \)[/tex] is the true annual interest rate
- [tex]\( c \)[/tex] is the loan amount
- [tex]\( m \)[/tex] is the number of monthly payments in a year
- [tex]\( n \)[/tex] is the total number of monthly payments
Given data:
- Loan amount ([tex]\( c \)[/tex]): [tex]$\$[/tex] 8,500[tex]$
- Monthly payments: $[/tex]\[tex]$ 170.50$[/tex]
- Time of loan contract ([tex]\( n \)[/tex]): 5 years
- True annual interest rate ([tex]\( y \)[/tex]): 10.0%
First, calculate the total number of monthly payments:
[tex]\[
m = 12
\][/tex]
[tex]\[
n = \text{time of the loan contract} = 5 \text{ years}
\][/tex]
[tex]\[
\text{Total number of monthly payments} = m \times n = 12 \times 5 = 60
\][/tex]
Now, plug these values into the formula to calculate the interest:
[tex]\[
I = \frac{2 \times y \times c}{m \times (n + 1)}
\][/tex]
Substituting the known values:
[tex]\[
y = 10.0\%
\][/tex]
[tex]\[
c = 8500
\][/tex]
[tex]\[
m = 12
\][/tex]
[tex]\[
n = 60
\][/tex]
[tex]\[
I = \frac{2 \times 10.0 \times 8500}{12 \times (60 + 1)}
\][/tex]
Simplify the denominator:
[tex]\[
12 \times 61 = 732
\][/tex]
Proceed with the numerator:
[tex]\[
2 \times 10.0 \times 8500 = 170000
\][/tex]
Now, divide the results:
[tex]\[
I = \frac{170000}{732} \approx 232.24043715846994
\][/tex]
Finally, the true annual interest rate, rounded to the nearest tenth, can be expressed as:
[tex]\[
I \approx 232.2
\][/tex]
Hence, the true annual interest rate is calculated as [tex]\( 232.2 \)[/tex].
