Suppose that you are thinking about buying a car and have narrowed down your choices to two options.
The​ new-car option: The new car costs ​$26 comma 000 and can be financed with a five​-year loan at 5.29​%.
The​ used-car option: A​ three-year old model of the same car costs ​$16 comma 000 and can be financed with a five​-year loan at 6.45​%.
What is the difference in monthly payments between financing the new car and financing the used​ car? Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFraction
.
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Part 1
The difference in monthly payments between financing the new car and financing the used car is ​$

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

To calculate the difference in monthly payments between financing the new car and financing the used car, we will use the formula for the monthly payment of a loan:

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

Where:
- $ P $ is the loan amount,
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of payments per year (12 for monthly payments),
- $ t $ is the loan term in years.

### For the new car:
- Loan amount $ P $ = $26,000
- Annual interest rate $ r $ = 5.29% = 0.0529
- Loan term $ t $ = 5 years
- $ n $ = 12 (monthly payments)

### For the used car:
- Loan amount $ P $ = $16,000
- Annual interest rate $ r $ = 6.45% = 0.0645
- Loan term $ t $ = 5 years
- $ n $ = 12 (monthly payments)

Let's calculate the monthly payments for both cars and then find the difference.# Defining the loan parameters
def calculate_monthly_payment(P, r, n, t):
    # Calculate the monthly interest rate
    monthly_interest_rate = r / n
    # Calculate the total number of payments
    total_payments = n * t
    # Using the PMT formula
    payment = (P * monthly_interest_rate) / (1 - (1 + monthly_interest_rate) ** -total_payments)
    return payment

# New car loan parameters
P_new_car = 26000  # Loan amount for the new car
r_new_car = 0.0529  # Annual interest rate for the new car
n = 12  # Number of payments per year
t = 5  # Loan term in years

# Used car loan parameters
P_used_car = 16000  # Loan amount for the used car
r_used_car = 0.0645  # Annual interest rate for the used car

# Calculate monthly payments for both new and used car
monthly_payment_new_car = calculate_monthly_payment(P_new_car, r_new_car, n, t)
monthly_payment_used_car = calculate_monthly_payment(P_used_car, r_used_car, n, t)

# Calculate the difference in monthly payments
monthly_payment_difference = monthly_payment_new_car - monthly_payment_used_car
monthly_payment_new_car, monthly_payment_used_car, monthly_payment_differenceThe monthly payment for the new car would be approximately $494.11, while the monthly payment for the used car would be approximately $312.68. 

The difference in monthly payments between financing the new car and the used car is about **$181.43**.

Would you like more details or have any other questions?

### Related Questions:
1. How does the total interest paid differ between the new car and the used car loans?
2. What is the total cost of each car after financing for 5 years?
3. How does the loan term affect the monthly payments and total interest?
4. What would be the monthly payments if the loan term were 6 years instead of 5 years?
5. What are the financial benefits of buying a used car instead of a new car?
6. How much would you need to pay as a down payment to reduce the new car's monthly payment by $50?
7. If the interest rate for the new car loan drops to 4.5%, how much would the monthly payment be?
8. How would an increase in the interest rate to 7% affect the used car's monthly payments?

**Tip**: Refinancing your loan could help lower your monthly payments if interest rates decrease after you purchase your car.

Learn how to calculate the monthly payment difference between financing a new car and a used car. Use the loan payment formula to compare costs based on loan amounts, interest rates, and loan terms. Get precise financial insights for making car purchasing decisions.

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