To determine the cash price of the car, we need to use the amortization formula for the loan. The formula for the present value (loan amount) of a series of monthly payments is:

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

Where:

• $PV$ is the present value (the loan amount).
• $P$ is the monthly payment ($110).
• $r$ is the monthly interest rate (annual interest rate divided by 12).
• $n$ is the total number of payments (36 in this case).

Step 1: Calculate the monthly interest rate

The annual interest rate is 6%, so the monthly interest rate is:

$r = \frac{6\%}{12} = \frac{0.06}{12} = 0.005$

Step 2: Calculate the loan amount

Now, plug the values into the amortization formula:

$PV = 110 \times \left( \frac{1 - (1 + 0.005)^{-36}}{0.005} \right)$

Let me calculate this.The loan amount, or the present value of the financed portion of the car, is approximately $3,615.81.

Step 3: Calculate the cash price of the car

The cash price of the car is the sum of the loan amount and the down payment of $500:

$\text{Cash Price} = 3615.81 + 500 = 4115.81$

Thus, the cash price of the car is approximately $4,115.81.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:

1. How would the cash price change if the interest rate were higher?
2. What happens to the monthly payments if the loan term is extended to 48 months?
3. How much total interest does Philip pay over the life of the loan?
4. What would be the cash price if Philip made a larger down payment?
5. How would the loan amount change if the interest were compounded annually instead?

Tip: Compounding frequency (monthly, quarterly, yearly) can significantly affect the overall cost of a loan due to interest accumulation.