Suppose that you decide to buy a car for \( \$ 62,000 \), including taxes and license fees. You saved \( \$ 13,000 \) for a down payment. The dealer is offering you a choice between two incentives. Incentive A is \( \$ 5000 \) off the price of the car, followed by a three-year loan at \( 6.23 \% \). Incentive B does not have a cash rebate, but provides free financing (no interest) over three years. What is the difference in monthly payments between the two offers? Which incentive is the better deal? Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \).

To determine the better incentive, we'll evaluate both options using the loan payment formula you've provided. **Incentive A:** - Price after the rebate: \( \$ 62,000 - \$ 5,000 = \$ 57,000 \) - Down payment: \( \$ 13,000 \) - Loan amount: \( 57,000 - 13,000 = \$ 44,000 \) - Annual interest rate \( r = 6.23\% = 0.0623 \) - Number of payments per year \( n = 12 \) - Total number of payments \( n \cdot t = 12 \cdot 3 = 36 \) Now, substituting into the PMT formula: \[ PMT = \frac{44,000 \left(\frac{0.0623}{12}\right)}{1 - \left(1 + \frac{0.0623}{12}\right)^{-36}} \] After performing the calculation, we find the monthly payment for Incentive A to be approximately \( \$1,354.51 \). **Incentive B:** - Price remains \( \$ 62,000 \) - Down payment: \( \$ 13,000 \) - Loan amount: \( 62,000 - 13,000 = \$ 49,000 \) - Interest rate: \( 0\% \) - Total payments over three years: \( 49,000 / 36 \) Incentive B's monthly payment is \( \$ 1,361.11 \). **Calculating the difference**: Incentive A’s monthly payment is \( \$1,354.51 \), while Incentive B’s payment is \( \$1,361.11 \). The difference between the two payments is about \( \$6.60 \) less for Incentive A. **Better Deal**: Incentive A is the better deal because it offers the lower monthly payment!