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Using the formula, compute the true annual interest rate.

[tex]\operatorname{Interest}(I)=\frac{2 y c}{m(n+1)}[/tex]

- Loan amount: \[tex]$8,500
- Monthly payments: \$[/tex]170.50
- Time of loan contract: 5 years

True annual interest rate (to the nearest tenth): [tex]\square \%[/tex]



Answer :

GinnyAnswer
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.

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