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 calculate Michael's monthly car payment for his $26,000 car loan at 4.25% compounded monthly, we will use the amortizing loan payment formula. Here’s a detailed step-by-step solution:

### Amortizing Loan Payment Formula
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]

Where:
- $ M $ = monthly payment
- $ P $ = principal loan amount ($26,000)
- $ r $ = monthly interest rate
- $ n $ = total number of monthly payments

### Step-by-Step Calculation

1. **Convert the annual interest rate to a monthly interest rate:**
   \[ r = \frac{4.25\%}{12} = \frac{0.0425}{12} \approx 0.00354167 \]

2. **Calculate the total number of monthly payments:**
   \[ n = 5 \text{ years} \times 12 \text{ months/year} = 60 \text{ months} \]

3. **Substitute the values into the formula:**
   \[ M = \frac{26000 \times 0.00354167 \times (1 + 0.00354167)^{60}}{(1 + 0.00354167)^{60} - 1} \]

4. **Calculate $(1 + r)^{60}$:**
   \[ (1 + 0.00354167)^{60} \approx 1.231448 \]

5. **Calculate the numerator:**
   \[ 26000 \times 0.00354167 \times 1.231448 \approx 113.5441 \]

6. **Calculate the denominator:**
   \[ (1 + 0.00354167)^{60} - 1 \approx 1.231448 - 1 = 0.231448 \]

7. **Calculate the monthly payment:**
   \[ M = \frac{113.5441}{0.231448} \approx 490.47 \]

### Final Answer

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 Michael's monthly car payment for a $26,000 loan at 4.25% interest compounded monthly over 5 years. Follow a detailed step-by-step solution using the amortizing loan payment formula. Suitable for advanced high school students and anyone interested in financial calculations.

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