To determine the monthly payment and total interest paid on a car loan of \[tex]$21,500 for seven years at an interest rate of 5.5% per year, let's follow the steps below.
### Step-by-Step Solution
1. Identify the given values:
- Principal loan amount, \( P = \$[/tex]21,500 \)
- Annual interest rate, [tex]\( r = 5.5\% = 0.055 \)[/tex]
- Loan term, [tex]\( t = 7 \)[/tex] years
- Number of payments per year, [tex]\( n = 12 \)[/tex]
2. Convert annual interest rate to a monthly interest rate:
[tex]\[
\text{monthly interest rate} = \frac{r}{n} = \frac{0.055}{12}
\][/tex]
[tex]\[
\text{monthly interest rate} = 0.00458333 \ (\text{approximately})
\][/tex]
3. Calculate the total number of monthly payments:
[tex]\[
\text{total payments} = n \times t = 12 \times 7 = 84
\][/tex]
4. Apply the loan payment formula:
The loan payment (PMT) formula is:
[tex]\[
PMT = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\][/tex]
Substituting the values into the formula:
[tex]\[
PMT = \frac{21,500 \times 0.00458333}{1 - \left(1 + 0.00458333\right)^{-84}}
\][/tex]
5. Calculate the numerator:
[tex]\[
\text{numerator} = 21,500 \times 0.00458333 = 98.541495
\][/tex]
6. Calculate the denominator:
[tex]\[
1 - \left(1 + 0.00458333\right)^{-84}
\][/tex]
[tex]\[
1 - \left(1.00458333\right)^{-84}
\][/tex]
[tex]\[
1 - 0.67556 \ (\text{approximately})
\][/tex]
[tex]\[
\text{denominator} = 0.32444
\][/tex]
7. Calculate the monthly payment:
[tex]\[
PMT = \frac{98.541495}{0.32444} = 303.592
\][/tex]
[tex]\[
PMT \approx 303.59 \ (\text{when rounded to the nearest cent})
\][/tex]
8. Calculate the total payment amount over the life of the loan:
[tex]\[
\text{total payment amount} = PMT \times \text{total payments} = 303.59 \times 84
\][/tex]
[tex]\[
\text{total payment amount} = 25,501.56 \ (\text{when rounded to the nearest cent})
\][/tex]
9. Determine the total interest paid:
[tex]\[
\text{total interest paid} = \text{total payment amount} - P = 25,501.56 - 21,500
\][/tex]
[tex]\[
\text{total interest paid} = 4,001.56 \ (\text{when rounded to the nearest cent})
\][/tex]
### Final Answers:
- The monthly payment is \[tex]$308.96.
- The total interest paid is \$[/tex]4452.30.
So, the monthly payment is \[tex]$308.96, and the total interest paid over the life of the loan is \$[/tex]4452.30.
