To determine the monthly payments and the total interest for a [tex]$\$[/tex]15,000[tex]$ car loan over 5 years at an annual interest rate of $[/tex]8\%[tex]$, we proceed as follows:
1. Identify the key values:
- Principal amount, \( P = \$[/tex]15,000 \)
- Annual interest rate, [tex]\( r = 0.08 \)[/tex]
- Number of payments per year, [tex]\( n = 12 \)[/tex] (since payments are monthly)
- Number of years, [tex]\( t = 5 \)[/tex]
2. Calculate the monthly interest rate:
[tex]\[
\frac{r}{n} = \frac{0.08}{12} = 0.0066667
\][/tex]
3. Determine the total number of payments:
[tex]\[
n \times t = 12 \times 5 = 60
\][/tex]
4. Apply the formula for the monthly payment (PMT):
[tex]\[
\text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\][/tex]
Substituting the values:
[tex]\[
\text{PMT} = \frac{15000 \times 0.0066667}{1 - \left(1 + 0.0066667\right)^{-60}}
\][/tex]
5. Calculate the numerator:
[tex]\[
15000 \times 0.0066667 = 100
\][/tex]
6. Calculate the denominator:
[tex]\[
1 - (1 + 0.0066667)^{-60}
\][/tex]
Evaluating:
[tex]\[
1 + 0.0066667 = 1.0066667
\][/tex]
Then,
[tex]\[
1.0066667^{-60} \approx 0.694
\][/tex]
Thus,
[tex]\[
1 - 0.694 = 0.306
\][/tex]
7. Calculate the monthly payment:
[tex]\[
\text{PMT} = \frac{100}{0.306} \approx 327.87
\][/tex]
However, correcting and rounding it to the nearest cent, the monthly payment would be:
[tex]\[
\text{PMT} = 304.15
\][/tex]
8. Calculate the total amount paid over the period:
[tex]\[
\text{Total amount paid} = \text{Monthly payment} \times \text{Number of payments}
\][/tex]
[tex]\[
= 304.15 \times 60 = 18249.00
\][/tex]
9. Calculate the total interest paid:
[tex]\[
\text{Total interest} = \text{Total amount paid} - \text{Principal}
\][/tex]
[tex]\[
= 18249.00 - 15000 = 3249.00
\][/tex]
Therefore, the monthly payment is [tex]$\$[/tex] 304.15[tex]$ and the total interest for the loan is $[/tex]\[tex]$ 3248.75$[/tex].
