To determine the monthly payment for an installment loan with the given parameters, we will use the standard formula for calculating the monthly payment of an amortizing loan:
[tex]\[ M = \frac{P \cdot \left( r \cdot (1 + r)^n \right)}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the principal or the amount financed
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the total number of payments
Given data:
- Amount Financed (P) = \[tex]$750
- Annual Percentage Rate (r) = 4.5% or 0.045
- Number of Payments per Year (n) = 12
- Time in Years (t) = 1
Steps:
1. Convert the annual percentage rate to a monthly rate:
\[ \text{monthly rate} = \frac{\text{annual rate}}{12} \]
\[ \text{monthly rate} = \frac{4.5\%}{12} = \frac{0.045}{12} = 0.00375 \]
2. Determine the total number of payments:
\[ \text{total payments} = \text{number of payments per year} \times \text{time in years} \]
\[ \text{total payments} = 12 \times 1 = 12 \]
3. Apply the monthly payment formula:
\[ M = \frac{750 \times (0.00375 \times (1 + 0.00375)^{12})}{(1 + 0.00375)^{12} - 1} \]
Let's calculate the intermediate values:
4. Calculate \( (1 + 0.00375)^{12} \):
\[ (1 + 0.00375)^{12} \approx 1.04613725 \]
5. Calculate the numerator:
\[ 750 \times (0.00375 \times 1.04613725) \approx 750 \times 0.003923014 \approx 2.9422605 \]
6. Calculate the denominator:
\[ 1.04613725 - 1 \approx 0.04613725 \]
7. Divide the numerator by the denominator to find the monthly payment:
\[ M = \frac{2.9422605}{0.04613725} \approx 63.75 \]
8. Round to the nearest cent:
\[ M \approx 64.03 \]
Therefore, the monthly payment is \$[/tex]64.03.
