To find the cash price of the car, we need to determine the total cost including both the down payment and the loan payments.

Step 1: Use the loan amortization formula to calculate the loan's present value.

The loan amortization formula for the present value of an annuity is given by:

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

Where:

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

Step 2: Identify the known values:

• Monthly payment, $P = 430$ dollars,
• Annual interest rate = 6% → monthly interest rate, $r = \frac{6\%}{12} = \frac{0.06}{12} = 0.005$,
• Number of payments, $n = 60$.

Step 3: Plug in the values:

$PV = 430 \times \frac{1 - (1 + 0.005)^{-60}}{0.005}$

$PV = 430 \times \frac{1 - (1.005)^{-60}}{0.005}$

$PV = 430 \times \frac{1 - 0.7408182207}{0.005}$

$PV = 430 \times \frac{0.2591817793}{0.005}$

$PV = 430 \times 51.83635586$

$PV \approx 22,289.63$

Step 4: Add the down payment to the loan's present value:

Lupé made a down payment of $8000. So, the total cash price of the car is:

$\text{Cash Price} = 22,289.63 + 8,000 = 30,289.63$

Final Answer:

The cash price of the car is $30,289.63.

Do you need more details or have any questions?

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Here are five related questions to expand on this topic:

1. How would the total price change if the interest rate were 7% instead of 6%?
2. What is the total amount of interest Lupé will pay over the life of the loan?
3. How would a shorter loan term (e.g., 48 months) affect the monthly payments?
4. What would be the impact if Lupé increased her down payment to $10,000?
5. How much would Lupé need to pay monthly if she wanted to pay off the loan in 36 months?

Tip: Always ensure you understand how interest compounding affects the total cost of a loan.