To determine the monthly payment for a loan of \[tex]$45,000 with an annual interest rate of 6.75%, to be paid off over five years, we use the given formula for monthly payments:
\[ M=\frac{P \left( \frac{r}{12} \right) \left(1+\frac{r}{12}\right)^n}{\left(1+\frac{r}{12}\right)^n-1} \]
Where:
- \( P \) is the amount borrowed (\$[/tex]45,000)
- [tex]\( r \)[/tex] is the annual interest rate (0.0675)
- [tex]\( n \)[/tex] is the total number of monthly payments (5 years × 12 months/year = 60 months)
Step-by-Step Solution:
1. Calculate Monthly Interest Rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{r}{12} = \frac{0.0675}{12} = 0.005625 \][/tex]
2. Identify Number of Payments:
[tex]\[ n = 5 \times 12 = 60 \][/tex]
3. Calculate the Numerator:
[tex]\[ \text{Numerator} = P \times \text{Monthly Interest Rate} \times (1 + \text{Monthly Interest Rate})^n \][/tex]
[tex]\[ = 45000 \times 0.005625 \times (1 + 0.005625)^{60} \][/tex]
[tex]\[ = 354.40409157180056 \][/tex]
4. Calculate the Denominator:
[tex]\[ \text{Denominator} = \left(1 + \frac{r}{12}\right)^n - 1 \][/tex]
[tex]\[ = (1 + 0.005625)^{60} - 1 \][/tex]
[tex]\[ = 0.4001149296663724 \][/tex]
5. Calculate Monthly Payment [tex]\( M \)[/tex]:
[tex]\[ M = \frac{\text{Numerator}}{\text{Denominator}} \][/tex]
[tex]\[ = \frac{354.40409157180056}{0.4001149296663724} \][/tex]
[tex]\[ = 885.7557299031884 \][/tex]
After rounding to two decimal places, the monthly payment [tex]\( M \)[/tex] is approximately 885.76. Therefore, the correct answer is:
(4) \$885.76
