Suppose that you decide to buy a car for 

​$59 comma 00059,000​, 

including taxes and license fees. You saved 

$ 10 comma 000$10,000 

for a down payment. The dealer is offering you a choice between two incentives.

Incentive A is 

​$60006000 

off the price of the​ car, followed by a 

fourfour​-year 

loan at 

6.656.65​%.

Incentive B does not have a cash​ rebate, but provides free financing​ (no interest) over 

fourfour 

years.

What is the difference in monthly payments between the two​ offers? Which incentive is the better​ deal? Use 

PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFractionPMT=Prn1−1+rn−nt.

Question content area bottom

Part 1

The difference in monthly payments between the two offers is 

​$enter your response here.

​(Round to the nearest cent as​ needed.)

To determine the difference in monthly payments between the two offers, we need to calculate the monthly payments under each incentive plan and then find the difference between them.

### **Step 1: Calculate the loan amount under each incentive.**

- **Incentive A:**
  - Car price: \$59,000
  - Cash rebate: \$6,000
  - Down payment: \$10,000
  - Loan amount under Incentive A: 
    \[
    59,000 - 6,000 - 10,000 = 43,000 \text{ dollars}
    \]

- **Incentive B:**
  - Car price: \$59,000
  - No rebate
  - Down payment: \$10,000
  - Loan amount under Incentive B: 
    \[
    59,000 - 10,000 = 49,000 \text{ dollars}
    \]

### **Step 2: Calculate the monthly payments using the PMT formula.**

The PMT formula is given as:
\[
\text{PMT} = \frac{P \times \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\]

Where:
- $ P $ = Loan amount
- $ r $ = Annual interest rate (as a decimal)
- $ n $ = Number of payments per year
- $ t $ = Number of years

Given:
- Annual interest rate for Incentive A, $ r = 6.65\% $ or $ 0.0665 $
- Number of years, $ t = 4 $ years
- Number of payments per year, $ n = 12 $ (monthly payments)

#### **Incentive A Monthly Payment:**

\[
\text{PMT}_A = \frac{43,000 \times \left(\frac{0.0665}{12}\right)}{1 - \left(1 + \frac{0.0665}{12}\right)^{-12 \times 4}}
\]

Let's calculate this value.

#### **Incentive B Monthly Payment:**

Incentive B has no interest, so the monthly payment is simply:

\[
\text{PMT}_B = \frac{49,000}{4 \times 12} = \frac{49,000}{48}
\]

Let's calculate these values to find the difference.

### **Step 3: Calculate the difference in monthly payments.**

Now, I'll calculate these values.### Results:
- **Monthly Payment under Incentive A:** \$1022.72
- **Monthly Payment under Incentive B:** \$1020.83
- **Difference in Monthly Payments:** \$1.89 (rounded to the nearest cent)

### Conclusion:
Incentive B offers a slightly lower monthly payment by \$1.89. Although the difference is minimal, Incentive B is technically the better deal in terms of monthly payments.

Would you like more details or have any questions?

### Related Questions:
1. How does the total interest paid compare between the two incentives?
2. What would the monthly payment be if the loan term were five years instead of four?
3. How would an increase in the interest rate to 7% affect the monthly payment under Incentive A?
4. What is the total cost difference over the loan term between the two incentives?
5. How would a down payment increase to \$15,000 impact the monthly payments?
6. What is the break-even point in total payments between the two incentives?
7. How would the monthly payment change under Incentive A if payments were made bi-weekly instead of monthly?
8. What would be the effect on monthly payments if the car price increased by \$5,000?

### Tip:
When comparing loan options, always consider both the interest rate and the loan term to determine the total cost over time, not just the monthly payment.

Learn how to calculate monthly payments and compare car loan incentives using the PMT formula. Solve for the difference in monthly payments between two car financing options. Suitable for advanced high school students studying finance and interest calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation