To calculate the monthly payment required to pay off a loan with a remaining principal of [tex]$1800 and an 8% annual interest rate, we need to follow these steps:
1. Convert the annual interest rate to a monthly interest rate:
The annual interest rate is given as 8%. To find the monthly interest rate, we divide the annual rate by 12 (the number of months in a year).
\[
\text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{12} = \frac{8\%}{12} = 0.08 / 12 = 0.006667
\]
So, the monthly interest rate is approximately 0.006667 (or 0.6667%).
2. Determine the number of monthly payments:
Since we are considering a period of one year to pay off the loan, the number of monthly payments is 12.
3. Use the formula for calculating the monthly payment on an installment loan:
The formula to calculate the monthly payment \( M \) is:
\[
M = P \frac{r (1 + r)^n}{(1 + r)^n - 1}
\]
where:
- \( P \) is the remaining principal ($[/tex]1800)
- [tex]\( r \)[/tex] is the monthly interest rate (0.006667)
- [tex]\( n \)[/tex] is the number of monthly payments (12)
4. Calculate the components of the formula:
- Calculate [tex]\((1 + r)^n\)[/tex]:
[tex]\[
(1 + 0.006667)^{12} \approx 1.082856
\][/tex]
- Calculate the numerator:
[tex]\[
\text{Numerator} = 1800 \times 0.006667 \times 1.082856 \approx 12.9922
\][/tex]
- Calculate the denominator:
[tex]\[
\text{Denominator} = 1.082856 - 1 = 0.082856
\][/tex]
5. Combine the components to find the monthly payment:
[tex]\[
M = \frac{\text{Numerator}}{\text{Denominator}} \approx \frac{12.9922}{0.082856} \approx 156.58
\][/tex]
Therefore, the final monthly payment required to pay off the loan is $156.58, rounded to the nearest cent.
