Let's solve the problem step-by-step using the given information:
1. Understand the given data:
- Current balance (principal, [tex]\( P \)[/tex]): \[tex]$18,875
- Annual interest rate: 11.6% (0.116 as a decimal)
- Number of months to pay off: 24
2. Convert the annual interest rate to a monthly interest rate:
- Monthly interest rate \( (i) \) = Annual interest rate / 12
- \( i = 0.116 / 12 \)
- \( i \approx 0.0096667 \)
3. Use the given formula for the monthly payment:
- The formula is:
\[
\text{Monthly Payment} = \frac{P \times i \times (1 + i)^{n}}{(1 + i)^{n} - 1}
\]
where:
- \( P \) = Principal amount (balance owed)
- \( i \) = Monthly interest rate
- \( n \) = Number of months
4. Substitute the known values into the formula:
- Principal, \( P = 18,875 \)
- Monthly interest rate, \( i \approx 0.0096667 \)
- Number of months, \( n = 24 \)
5. Calculate the numerator and denominator separately:
\[
\text{Numerator} = P \times i \times (1 + i)^{n}
\]
\[
\text{Denominator} = (1 + i)^{n} - 1
\]
6. Plug the values into the equations:
- Numerator:
\[
\text{Numerator} = 18,875 \times 0.0096667 \times (1 + 0.0096667)^{24} \approx 229.85
\]
- Denominator:
\[
\text{Denominator} = (1 + 0.0096667)^{24} - 1 \approx 0.2597
\]
7. Divide the numerator by the denominator to find the monthly payment:
\[
\text{Monthly Payment} = \frac{229.85}{0.2597} \approx 884.99
\]
Seth's approximate monthly payment will be \$[/tex]884.99.
Given the options:
- A. \[tex]$834.75
- B. \$[/tex]884.99
- C. \[tex]$868.35
- D. \$[/tex]753.97
The correct answer is B. Seth's approximate monthly payment will be \$884.99.
