Michael takes out a $26,000 car loan at 4.25% compounded monthly to purchase a sport utility vehicle. If he plans on paying off the loan in five years, what is his monthly car payment?

To find the monthly car payment for Michael's car loan, we need to use the formula for the monthly payment on an amortizing loan, which is given by:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the principal amount of the loan, which is $26,000.
- $ r $ is the monthly interest rate, which is the annual interest rate divided by 12. So, $ r = \frac{4.25\%}{12} = \frac{0.0425}{12} $.
- $ n $ is the number of monthly payments, which is the number of years multiplied by 12. So, $ n = 5 \times 12 = 60 $.

First, we calculate the monthly interest rate:

\[ r = \frac{0.0425}{12} \approx 0.00354167 \]

Next, we calculate the number of payments:

\[ n = 5 \times 12 = 60 \]

Now we plug these values into the formula:

\[ M = \frac{26000 \cdot 0.00354167 \cdot (1 + 0.00354167)^{60}}{(1 + 0.00354167)^{60} - 1} \]

Let's break it down step by step:

1. Calculate $(1 + r)^{60}$:

\[ (1 + 0.00354167)^{60} \approx 1.231448 \]

2. Calculate the numerator:

\[ 26000 \cdot 0.00354167 \cdot 1.231448 \approx 113.5441 \]

3. Calculate the denominator:

\[ (1 + 0.00354167)^{60} - 1 \approx 1.231448 - 1 = 0.231448 \]

4. Divide the numerator by the denominator:

\[ M = \frac{113.5441}{0.231448} \approx 490.47 \]

Therefore, Michael's monthly car payment is approximately $490.47.

Do you have any questions or would you like more details on any part of this solution?

Here are five numbered questions you might consider asking next:
1. How is the monthly interest rate derived from the annual interest rate?
2. What is the significance of the amortization formula in loan payments?
3. How would the monthly payment change if the loan term was extended to 6 years?
4. What impact does increasing the interest rate have on the monthly payment?
5. How can I calculate the total interest paid over the life of the loan?

**Tip:** Always ensure that the interest rate and the number of periods are in the same units (e.g., monthly) when using the loan payment formula.

Learn how to calculate the monthly car payment for a $26,000 car loan at 4.25% annual interest rate compounded monthly. This problem covers interest calculations, compound interest, and loan amortization. Suitable for advanced high school students and college-level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation