Paula is considering the purchase of a new car. She has narrowed her search to two cars that are equally appealing to her. Car A costs $29,000, and Car B costs $29,400. The manufacturer of Car A is offering 0% financing for 48 months with zero down, while the manufacturer of Car B is offering a rebate of $2000 at the time of purchase plus financing at the rate of 3%/year compounded monthly over 48 months with zero down. If Paula has decided to buy the car with the lower net cost to her, which car should she purchase? (Round numerical values to the nearest cent.)

We will compare the total cost of both Car A and Car B over the 48-month period to determine which option is more cost-effective for Paula.

### Step 1: **Cost of Car A**
- Car A costs \$29,000.
- 0% financing over 48 months means Paula will pay off the \$29,000 in 48 equal monthly installments with no interest.

The total cost for Car A is simply the purchase price:
\[
\text{Total cost of Car A} = \$29,000
\]

### Step 2: **Cost of Car B**
- Car B has a price of \$29,400, but there is a \$2000 rebate, so the initial cost is reduced to:
\[
\text{Initial cost of Car B} = 29,400 - 2000 = 27,400
\]
- Financing for Car B is at 3% per year, compounded monthly for 48 months. To calculate the monthly payments, we use the formula for the monthly payment of a loan:

\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where:
- $ M $ is the monthly payment,
- $ P = 27,400 $ is the loan amount,
- $ r = \frac{3\%}{12} = \frac{0.03}{12} = 0.0025 $ is the monthly interest rate,
- $ n = 48 $ is the number of months.

Now, we can calculate the monthly payment.

\[
M = 27,400 \times \frac{0.0025(1 + 0.0025)^{48}}{(1 + 0.0025)^{48} - 1}
\]

Let me calculate this.The monthly payment for Car B is approximately \$606.48. Over 48 months, the total cost of Car B is:

\[
\text{Total cost of Car B} = 606.48 \times 48 = 29,111.07
\]

### Step 3: **Comparison**
- Total cost of Car A: \$29,000
- Total cost of Car B: \$29,111.07

Since the total cost of Car A (\$29,000) is lower than the total cost of Car B (\$29,111.07), Paula should purchase **Car A**.

Would you like more details, or have any questions?

Here are 5 related questions to consider:
1. How does the length of the loan period affect the total interest paid?
2. What if Car B had a lower interest rate—how would that impact the total cost?
3. How would the calculations change if Paula made a down payment on either car?
4. What is the impact of compounding frequency on loan interest?
5. How can Paula further minimize financing costs?

**Tip:** When considering financing options, always compare the total amount paid over time, not just the sticker price or monthly payment.

Paula is considering the purchase of a new car. She has narrowed her search to two cars that are equally appealing to her. Car A costs $29,000, and Car B costs $29,400. The manufacturer of Car A is offering 0% financing for 48 months with zero down, while the manufacturer of Car B is offering a rebate of $2000 at the time of purchase plus financing at the rate of 3%/year compounded monthly over 48 months with zero down. If Paula has decided to buy the car with the lower net cost to her, which car should she purchase?

Learn how to compare car financing options by calculating total loan costs. This example contrasts 0% financing with a rebate and 3% annual interest, compounded monthly. Detailed step-by-step calculations make this problem suitable for high school students learning about loan payments and interest rates.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation