To calculate the difference in monthly payments between the new car and the used car, we will use the formula you provided for monthly payments on a loan:

$PMT = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}$

Where:

• $P$ = loan principal (amount borrowed)
• $r$ = annual interest rate (as a decimal)
• $n$ = number of payments per year (monthly payments, so $n = 12$)
• $t$ = loan term in years

For the New Car:

• Principal $P = 31,000$
• Annual interest rate $r = 5.84\% = 0.0584$
• Loan term $t = 4$ years
• $n = 12$ payments per year

For the Used Car:

• Principal $P = 18,000$
• Annual interest rate $r = 7.61\% = 0.0761$
• Loan term $t = 4$ years
• $n = 12$ payments per year

We will calculate the monthly payments for both cars and then find the difference.

Step 1: Calculate the monthly payment for the new car

$PMT_{\text{new}} = \frac{31,000 \left( \frac{0.0584}{12} \right)}{1 - \left( 1 + \frac{0.0584}{12} \right)^{-12 \times 4}}$

Step 2: Calculate the monthly payment for the used car

$PMT_{\text{used}} = \frac{18,000 \left( \frac{0.0761}{12} \right)}{1 - \left( 1 + \frac{0.0761}{12} \right)^{-12 \times 4}}$

Let's perform these calculations now to determine the difference. Given values P_new = 31000 # Principal for new car P_used = 18000 # Principal for used car r_new = 0.0584 # Interest rate for new car r_used = 0.0761 # Interest rate for used car n = 12 # Monthly payments per year t = 4 # Loan term in years

Calculate monthly payments for the new car

PMT_new = (P_new * (r_new / n)) / (1 - (1 + r_new / n) ** (-n * t))

Calculate monthly payments for the used car

PMT_used = (P_used * (r_used / n)) / (1 - (1 + r_used / n) ** (-n * t))

Calculate the difference in monthly payments

difference = PMT_used - PMT_new

PMT_new, PMT_used, differenceThe monthly payment for the new car is approximately $725.76, while the monthly payment for the used car is approximately $436.14.

The difference in monthly payments between financing the new car and financing the used car is approximately $289.62.

This means the new car has a higher monthly payment by $289.62.

Let me know if you'd like further details or have any questions!

Here are 5 related questions:

1. How do loan interest rates affect monthly payments?
2. What is the total amount paid for the new car over the course of the loan?
3. How do the loan terms (e.g., 4 years vs. 5 years) impact the monthly payments?
4. Can you compare the total interest paid on both loans?
5. What are the advantages and disadvantages of financing a new vs. a used car?

Tip: Even though the new car has a higher monthly payment, consider the long-term value, warranty, and maintenance costs when making a decision.