Determine the monthly payment for the installment loan.

\begin{tabular}{|c|c|c|c|}
\hline
\begin{tabular}{l}
Amount \\
Financed (P)
\end{tabular} & \begin{tabular}{l}
Annual \\
Percentage \\
Rate (r)
\end{tabular} & \begin{tabular}{l}
Number of \\
Payments per \\
Year (n)
\end{tabular} & \begin{tabular}{l}
Time in \\
Years (t)
\end{tabular} \\
\hline
\[tex]$ 15,000 & 5.5\% & 12 & 3 \\
\hline
\end{tabular}

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The monthly payment is \$[/tex] [tex]$\square$[/tex]
(Round to the nearest cent as needed.)



Answer :

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.