Suppose that you decide to buy a car for $30,635, including taxes and license fees. You saved $9000 for a down payment and can get a three-year car loan at 6.77%. Use PMT=P(r/n)/[1-(1+r/n)^nt] to find the monthly payment and the total interest for the loan. Round to the nearest cent as needed.

We are given the following information to calculate the monthly payment for the car loan:

- **Car price (including taxes and fees):** $30,635
- **Down payment:** $9,000
- **Loan interest rate:** 6.77% per year
- **Loan term:** 3 years (36 months)

### Step 1: Calculate the loan amount
First, subtract the down payment from the total car price to get the loan amount:

\[
\text{Loan Amount} = 30,635 - 9,000 = 21,635
\]

### Step 2: Define the terms in the formula

We are using the formula for monthly payments:

\[
PMT = P \times \frac{r/n}{1 - (1 + r/n)^{-nt}}
\]

where:
- $ PMT $ = monthly payment
- $ P $ = loan amount = $21,635
- $ r $ = annual interest rate = 6.77% = 0.0677
- $ n $ = number of payments per year = 12 (since we are calculating monthly payments)
- $ t $ = loan term in years = 3

### Step 3: Plug the values into the formula

Now substitute the values into the formula:

\[
PMT = 21,635 \times \frac{0.0677/12}{1 - (1 + 0.0677/12)^{-12 \times 3}}
\]

I'll calculate this step-by-step.The monthly payment for the car loan is approximately **$665.75**.

### Step 4: Calculate the total interest paid

Now, let's find the total interest paid over the life of the loan. First, we calculate the total amount paid over the 36 months, then subtract the original loan amount:

\[
\text{Total amount paid} = PMT \times \text{Total number of payments}
\]
\[
\text{Total interest} = (\text{Total amount paid}) - P
\]

Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the total interest by following these steps:

1. Multiply the monthly payment (\$665.75) by the total number of payments (36 months).
   \[
   \text{Total amount paid} = 665.75 \times 36 = 23,967
   \]

2. Subtract the loan amount (\$21,635) from the total amount paid.
   \[
   \text{Total interest} = 23,967 - 21,635 = 2,332
   \]

So, the total interest paid over the loan would be approximately **$2,332**.

Would you like further details or have any other questions?

### Related Questions:
1. How does the interest rate affect monthly payments?
2. What happens if you change the loan term to 5 years instead of 3?
3. How can you reduce the total interest paid over the loan period?
4. What would the monthly payment be if the down payment were $5,000 instead of $9,000?
5. How does the type of loan (fixed vs. variable rate) affect the total cost of the loan?

**Tip:** Making extra payments toward the loan principal can significantly reduce the total interest paid over the life of the loan.

Learn how to calculate the monthly payment and total interest for a car loan with a 6.77% interest rate. This solution involves key concepts like loan amortization, algebra, and interest rate calculations. Suitable for students in Grades 10-12.

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