You have just taken out a $ 22 comma 000 car loan with a 8 % ​APR, compounded monthly.  The loan is for five years. When you make your first payment in one​ month, how much of the payment will go toward the principal of the loan and how much will go toward​ interest?  ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.) 
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Part 1
When you make your first​ payment, ​$
  
enter your response here will go toward the principal of the loan and ​$
  
enter your response here will go toward the interest.  ​(Round to the nearest​ cent.)

To solve this, we need to calculate the monthly payment using the formula for a fixed-rate loan payment:

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

where:
- $ P $ = principal amount (loan amount) = $22,000
- $ r $ = monthly interest rate = $\frac{8\%}{12}$ = $0.08 / 12$
- $ n $ = total number of payments = 5 years × 12 months = 60

### Step 1: Calculate the monthly interest rate
\[
r = \frac{0.08}{12} = 0.00666667
\]

### Step 2: Calculate the total number of payments
\[
n = 5 \times 12 = 60
\]

### Step 3: Calculate the monthly payment
Substitute $ P $, $ r $, and $ n $ into the monthly payment formula:

\[
M = 22000 \times \frac{0.00666667 \times (1 + 0.00666667)^{60}}{(1 + 0.00666667)^{60} - 1}
\]

Calculating this step-by-step:

1. $ (1 + 0.00666667)^{60} \approx 1.489845 $
2. $ 0.00666667 \times 1.489845 \approx 0.0099323 $
3. The denominator is $ 1.489845 - 1 = 0.489845 $

Thus,

\[
M = 22000 \times \frac{0.0099323}{0.489845} \approx 446.88
\]

So, the monthly payment is approximately $446.88.

### Step 4: Calculate the interest portion of the first payment
The interest for the first month is calculated as:

\[
\text{Interest} = P \times r = 22000 \times 0.00666667 = 146.67
\]

### Step 5: Calculate the principal portion of the first payment
The principal portion is:

\[
\text{Principal} = M - \text{Interest} = 446.88 - 146.67 = 300.21
\]

### Final Answer:
- The amount going toward the principal is **$300.21**.
- The amount going toward the interest is **$146.67**.

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. What would happen if the loan was for 10 years instead of 5?
2. How does the payment change if the interest rate decreases to 6% APR?
3. What is the total amount paid over the life of the loan?
4. How much of the total payment goes toward interest over the 5-year period?
5. What would the payments be if the interest were compounded daily instead of monthly?

**Tip:** Always double-check interest rate conversions when calculating loan payments (e.g., converting annual rate to monthly).

You have just taken out a $22,000 car loan with a 8% APR, compounded monthly. The loan is for five years. When you make your first payment in one month, how much of the payment will go toward the principal of the loan and how much will go toward interest? (Note: Be careful not to round any intermediate steps less than six decimal places.)

Learn how to calculate the first payment on a $22,000 car loan with 8% APR, compounded monthly, over 5 years. This step-by-step solution covers the loan payment formula and the division of payments into principal and interest. Suitable for high school and college students.

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