Harper is considering the purchase of a ​$9 comma 298 used car and is trying to decide between a 48​-month and a 60​-month car loan. If the loan is for ​$9 comma 298 at 5 ​percent, what is the difference in the monthly​ payments?
​Note: Round intermediate computations to at least five​ (5) decimal places.
Click on the table icon to view the Monthly Installment Loan Payment Factor​ (MILPF) table: LOADING....
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Part 1
The monthly payment for the 48​-month car loan is ​$

enter your response here.  ​(Round to the nearest​ cent.)
Part 2
The monthly payment for the 60​-month car loan is ​$

enter your response here.  ​(Round to the nearest​ cent.)
Part 3
The difference between the monthly payments is ​$

enter your response here.  ​(Round to the nearest​ cent.)

To solve this, we use the loan payment formula for fixed monthly payments, which is derived from the **amortization formula**:

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

Where:
- $M$ is the monthly payment
- $P$ is the principal loan amount (\$9,298)
- $r$ is the monthly interest rate (annual rate of 5% = 0.05, divided by 12 months)
- $n$ is the total number of monthly payments (48 or 60 months in this case)

### Part 1: Monthly Payment for a 48-month Loan

For the 48-month loan:
- $P = 9,298$
- $r = \frac{0.05}{12} = 0.0041667$
- $n = 48$

Substitute these values into the formula:

\[
M = \frac{9,298 \times 0.0041667 \times (1 + 0.0041667)^{48}}{(1 + 0.0041667)^{48} - 1}
\]

Let's calculate this.

### Part 2: Monthly Payment for a 60-month Loan

For the 60-month loan:
- $P = 9,298$
- $r = 0.0041667$
- $n = 60$

Substitute these values into the formula:

\[
M = \frac{9,298 \times 0.0041667 \times (1 + 0.0041667)^{60}}{(1 + 0.0041667)^{60} - 1}
\]

We'll compute this next.

### Part 3: Difference Between Monthly Payments

Once we compute both payments, the difference is simply:

\[
\text{Difference} = \text{Monthly payment for 48 months} - \text{Monthly payment for 60 months}
\]

Let me do the calculations to give you the precise values.### Results:

1. **Monthly payment for the 48-month loan**: \$214.13
2. **Monthly payment for the 60-month loan**: \$175.46
3. **Difference between the monthly payments**: \$38.66

This means the 48-month loan will cost \$38.66 more per month compared to the 60-month loan.

Would you like further details, or do you have any questions?

Here are some related questions:
1. What factors should Harper consider when choosing between a 48-month and a 60-month loan?
2. How does extending the loan term affect the total interest paid over time?
3. What are the pros and cons of lower monthly payments with longer loan terms?
4. How does the interest rate impact the loan's monthly payments?
5. How would Harper's decision change if the interest rate was higher?

**Tip**: The longer the loan term, the more interest you’ll pay overall, even though the monthly payment is lower.

Harper is considering the purchase of a $9,298 used car and is trying to decide between a 48-month and a 60-month car loan. If the loan is for $9,298 at 5 percent, what is the difference in the monthly payments?

Discover how to calculate monthly payments for a $9,298 car loan with a 5% interest rate over 48 and 60 months. This solution explains the loan amortization formula, interest calculation, and the difference in monthly payments for different loan terms. Ideal for high school students in Grades 10-12.

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