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Suppose that you decide to buy a car for $57,000, including taxes and license fees. You saved $12,000 for a down payment. The dealer is offering you a choice between two incentives:


  • Incentive A is $5,000 off the price of the car, followed by a four-year loan at 7.27%.
  • Incentive B does not have a cash rebate, but provides free financing (no interest) over four years.



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



$PMT=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]}$

Verified Answer :

  • Principal for A: \(P_A = \$40,000\); Principal for B: \(P_B = \$45,000\).
  • Monthly payment A: \(PMT_A \approx \$962.87\); Monthly payment B: \(PMT_B = \$937.50\).
  • The difference in monthly payments is \(\boxed{\$25.37}\).
  • Total costs: Incentive A costs \(\$58,217.68\), while Incentive B costs \(\$57,000\); Incentive B is the better deal.

Explanation

  1. Analysis and Problem Setup

We need to compare two financing options for a car purchase with a total price of \(\$57,000\) and a \(\$12,000\) down payment.

  • The loan principal for Incentive A is the total price minus the rebate and the down payment: \(P_A = 57,000 - 5,000 - 12,000 = 40,000\). The loan terms are 4 years (48 months) at an annual interest rate of \(7.27\%\).
  • The loan principal for Incentive B is the total price minus the down payment: \(P_B = 57,000 - 12,000 = 45,000\). The loan terms are 4 years (48 months) at \(0\%\) interest.
  1. Calculating Monthly Payments and Difference

Using the formula \(PMT = \frac{P(\frac{r}{n})}{[1 - (1 + \frac{r}{n})^{-nt}]}\), we calculate the monthly payment for Incentive A with \(P_A = 40,000\), \(r = 0.0727\), \(n = 12\), and \(t = 4\):
\[PMT_A = \frac{40000(\frac{0.0727}{12})}{[1 - (1 + \frac{0.0727}{12})^{-48}]} \approx \frac{40000 \times 0.006058333}{1 - (1.006058333)^{-48}} \approx \frac{242.3333}{1 - 0.748366} \approx 962.87\]
For Incentive B, with \(0\%\) interest, the payment is simply the principal divided by the number of months:
\[PMT_B = \frac{45000}{48} = 937.50\]
The difference in monthly payments is \(|962.87 - 937.50| = 25.37\).

  1. Comparing Total Costs and Conclusion

Now we compare the total cost of each incentive.

  • For Incentive A, the total cost is: \((\text{Monthly Payment} \times \text{Number of Months}) + \text{Down Payment}\).

\[\text{Total Cost}_A = (962.87 \times 48) + 12000 \approx 46217.68 + 12000 = 58217.68\]

  • For Incentive B, the total cost is:

\[\text{Total Cost}_B = (937.50 \times 48) + 12000 = 45000 + 12000 = 57000\]
Incentive B is the better deal because it results in a lower total cost (\(\$57,000\) versus \(\$58,217.68\)).

Examples

This scenario demonstrates how to analyze financing options by comparing monthly payments and total costs. In real life, dealerships often offer choices between cash discounts and low-interest financing. Understanding these calculations helps consumers make informed decisions, ensuring they identify the option that minimizes their total expenditure over the life of a loan. Similar logic applies to mortgages, personal loans, or even credit card debt management.

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