Lupé made a down payment of $8000 toward the purchase of a new car. To pay the balance of the purchase price, she has secured a loan from her bank at the rate of 6%/year compounded monthly. Under the terms of her finance agreement she is required to make payments of $430/month for 60 months. What is the cash price of the car? (Round your answer to the nearest cent.)


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.

Lupé made a down payment of $8000 toward the purchase of a new car. To pay the balance of the purchase price, she has secured a loan from her bank at the rate of 6%/year compounded monthly. Under the terms of her finance agreement, she is required to make payments of $430/month for 60 months. What is the cash price of the car? (Round your answer to the nearest cent.)

Learn how to calculate the total cash price of a car by considering a down payment and loan payments. This example includes calculating the present value of a loan with 6% annual interest, compounded monthly, and payments of $430 per month over 60 months. Ideal for students in Grades 11-12 and finance enthusiasts.

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