To determine the monthly payment for the installment loan, we need to follow a series of steps and apply the relevant formula. Here’s the step-by-step solution:
1. Identify the given information:
- Amount Financed ([tex]\(P\)[/tex]) = [tex]$15,000
- Annual Percentage Rate (APR, \(r_{\text{annual}}\)) = 5.5%
- Number of Payments per Year (\(n\)) = 12
- Time in Years (\(t\)) = 3
2. Convert the annual interest rate to a monthly interest rate:
The monthly interest rate (\(r_{\text{monthly}}\)) can be calculated by dividing the annual interest rate by the number of payments per year.
\[
r_{\text{monthly}} = \frac{r_{\text{annual}}}{n} = \frac{0.055}{12} = 0.004583333333333333
\]
3. Calculate the total number of payments over the entire loan period:
\[
\text{Total Payments} = n \times t = 12 \times 3 = 36
\]
4. Use the formula for the monthly payment of an installment loan:
The formula is:
\[
M = P \times \frac{r_{\text{monthly}} (1 + r_{\text{monthly}})^{\text{Total Payments}}}{(1 + r_{\text{monthly}})^{\text{Total Payments}} - 1}
\]
Substituting in the given values:
\[
M = 15000 \times \frac{0.004583333333333333 (1 + 0.004583333333333333)^{36}}{(1 + 0.004583333333333333)^{36} - 1}
\]
5. Calculate the monthly payment:
\[
M \approx 452.94
\]
6. Round the result to the nearest cent:
Thus, the monthly payment for the installment loan is \$[/tex]452.94.
