You would like to purchase a car with a negotiated price of $30,000, and the dealer offers financing over a five-year period at 8%. If repayments are to be made annually, what would be your annual payments?

To calculate the annual payments for the car loan, we can use the formula for the **annual payment of an amortizing loan**, which is given by:

\[
A = \frac{P \cdot r}{1 - (1 + r)^{-n}}
\]

Where:
- $ A $ = Annual payment
- $ P $ = Principal (loan amount) = \$30,000
- $ r $ = Annual interest rate = 8% = 0.08
- $ n $ = Number of payments (years) = 5

Now, let's plug the values into the formula:

\[
A = \frac{30,000 \cdot 0.08}{1 - (1 + 0.08)^{-5}}
\]

First, calculate $ (1 + 0.08)^{-5} $:

\[
(1 + 0.08)^{-5} = 1.08^{-5} \approx 0.6806
\]

Now, calculate the denominator:

\[
1 - 0.6806 = 0.3194
\]

Next, calculate the numerator:

\[
30,000 \cdot 0.08 = 2,400
\]

Finally, calculate the annual payment:

\[
A = \frac{2,400}{0.3194} \approx 7,515.67
\]

Therefore, your **annual payment** would be approximately **\$7,515.67**.

Would you like more details on this calculation, or do you have any questions?

Here are 5 related questions to expand on this:
1. What if the loan period was 7 years instead of 5?
2. How much total interest would you pay over the 5 years?
3. What would the payments be if the interest rate was 6% instead of 8%?
4. What is the effect of making monthly payments instead of annual ones?
5. How does the interest rate affect the overall cost of the loan?

**Tip:** Always check if making larger payments or extra payments will help you reduce the total interest paid.

This problem demonstrates how to calculate the annual payments for a $30,000 car loan with 8% interest over a 5-year period. Learn to apply the amortizing loan formula and explore related finance concepts. Suitable for students in Grades 11-12 or college-level finance courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation