The salesperson offers, "Buy this new car for $25,000 cash or, with appropriate down payment, pay $500 per month for 48 months at 8% interest." Assuming that the salesperson does not offer a free lunch, calculate the "appropriate" down payment.

To calculate the appropriate down payment, we'll break down the problem into a few steps:

### Step 1: Calculate the present value of the loan payments
The monthly payment is $500, and the loan term is 48 months. The interest rate is 8% annually, but we need to use the monthly interest rate for our calculations.

The monthly interest rate is:

\[
\text{Monthly Interest Rate} = \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667
\]

Now, we'll use the formula for the present value of an annuity to calculate the total value of the 48 payments:

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

Where:
- $ PV $ is the present value of the loan (amount to be financed),
- $ P $ is the monthly payment,
- $ r $ is the monthly interest rate,
- $ n $ is the number of payments.

Substituting the values:

\[
PV = 500 \times \frac{1 - (1 + 0.0066667)^{-48}}{0.0066667}
\]

Let's compute this value.The present value of the loan payments is approximately $20,480.96.

### Step 2: Calculate the appropriate down payment
The total cost of the car is $25,000. Since the present value of the loan is $20,480.96, the down payment would be:

\[
\text{Down Payment} = \text{Total Cost} - \text{Present Value of Loan}
\]

\[
\text{Down Payment} = 25,000 - 20,480.96 \approx 4,519.04
\]

### Final Answer:
The appropriate down payment is approximately **$4,519.04**.

Would you like more details or have any questions?

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Here are some related questions:
1. How is the present value of an annuity calculated in different interest scenarios?
2. What if the interest rate was compounded quarterly instead of monthly?
3. How does the length of the loan term affect the required down payment?
4. What happens if the monthly payment is increased or decreased?
5. How do different types of loans (fixed vs. variable interest rates) affect the present value calculation?
6. What if there was a balloon payment at the end of the loan term?
7. How would a zero-interest loan affect the down payment?
8. Can the present value formula be used for other types of recurring payments?

**Tip:** When dealing with loan calculations, always check whether the interest rate provided is annual, and if so, convert it to the appropriate period (monthly, quarterly) before calculating.

Learn how to calculate the appropriate down payment for purchasing a car using a loan, based on monthly payments, interest rate, and total cost. This problem involves applying the formula for the present value of an annuity. Suitable for finance students and anyone interested in understanding loan calculations.

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