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 9%/year compounded monthly. Under the terms of her finance agreement she is required to make payments of $460/month for 48 months. What is the cash price of the car? (Round your answer to the nearest cent.)


To determine the cash price of the car, we need to find the total loan amount Lupé secured from the bank, then add her down payment to that loan amount.

Here's a step-by-step breakdown:

### Step 1: Calculate the Loan Amount
Lupé is making monthly payments for a total of 48 months. The loan is compounded monthly at an interest rate of 9% per year, which means:

- Monthly interest rate, $ i = \frac{9\%}{12} = 0.75\% = 0.0075 $
- Monthly payment, $ P = 460 $
- Number of payments, $ n = 48 $

We can use the formula for the present value of an ordinary annuity to calculate the loan amount $ A $:

\[
A = P \times \left( \frac{1 - (1 + i)^{-n}}{i} \right)
\]

Substituting the known values:

\[
A = 460 \times \left( \frac{1 - (1 + 0.0075)^{-48}}{0.0075} \right)
\]

### Step 2: Compute the Loan Amount
First, calculate $ (1 + 0.0075)^{-48} $:

\[
1 + 0.0075 = 1.0075
\]
\[
1.0075^{-48} \approx 0.69218
\]

Now substitute into the equation:

\[
A = 460 \times \left( \frac{1 - 0.69218}{0.0075} \right)
\]
\[
A = 460 \times \left( \frac{0.30782}{0.0075} \right)
\]
\[
A \approx 460 \times 41.043
\]
\[
A \approx 18,879.78
\]

This is the loan amount Lupé secured from the bank.

### Step 3: Calculate the Cash Price of the Car
Now, add the down payment of $8,000 to the loan amount to find the total cash price of the car:

\[
\text{Cash price} = 18,879.78 + 8,000 = 26,879.78
\]

Thus, the cash price of the car is **$26,879.78**.

---

Would you like more details about the steps, or have any questions? Here are some related questions to explore:

1. What would be the monthly payment if the interest rate were 8% instead of 9%?
2. How would the total amount paid for the car change if the loan duration were 60 months instead of 48?
3. How is the present value of an annuity calculated, and why is it important in loan calculations?
4. How much would Lupé have saved in interest if she had made a down payment of $10,000 instead of $8,000?
5. What is the difference between compound interest and simple interest in loan calculations?

**Tip:** Always check if you are dealing with compound or simple interest when calculating loan payments, as this significantly affects the total cost.

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 9%/year compounded monthly. Under the terms of her finance agreement she is required to make payments of $460/month for 48 months. What is the cash price of the car? (Round your answer to the nearest cent.)

Learn how to calculate the cash price of a car with a detailed example involving a loan of 9% annual interest compounded monthly and monthly payments of $460 over 48 months. This example includes annuity formulas and practical finance concepts, suitable for students in introductory finance courses.

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