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Suppose you decide to buy a car for [tex]$59,000, including taxes and license fees. You have saved $[/tex]13,000 for a down payment. The dealer offers two incentives:

Incentive A: [tex]$6,000 off the price of the car, followed by a three-year loan at 6.75%.

\ \textless \ strong\ \textgreater \ Incentive B\ \textless \ /strong\ \textgreater \ : No 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 the formula:

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

The difference in monthly payments between the two offers is $[/tex] ______.
(Round to the nearest cent as needed.)

Sagot :

GinnyAnswer
To determine the difference in monthly payments between Incentive A and Incentive B, let's go through the calculations step-by-step:

### 1. Definitions and Initial Values
- Car Price: \[tex]$59,000 - Down Payment: \$[/tex]13,000

### 2. Incentive A
- Rebate Amount: \[tex]$6,000 - Loan Term: 3 years (36 months) - Annual Interest Rate: 6.75% (0.0675 as a decimal) First, calculate the loan amount for Incentive A: - Loan Amount for Incentive A: \((\text{Car Price} - \text{Rebate Amount}) - \text{Down Payment}\) - \(\$[/tex]59,000 - \[tex]$6,000 - \$[/tex]13,000 = \[tex]$40,000\) Next, calculate the monthly interest rate for Incentive A: - Monthly Interest Rate: \(\frac{0.0675}{12}\) We will use the monthly payment formula for loans: \[ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right]} \] Where: - \( P \) = Loan Amount - \( r \) = Annual Interest Rate - \( n \) = Number of payments per year (12 for monthly) - \( t \) = Loan term in years Substitute the known values: - \( P = 40,000 \) - \( r = 0.0675 \) - \( n = 12 \) - \( t = 3 \) ### 3. Incentive B - No rebate is offered, but free financing is provided (0% interest). - Loan Amount for Incentive B: \(\$[/tex]59,000 - \[tex]$13,000 = \$[/tex]46,000\)
- Loan Term: 3 years (36 months)

Since there's no interest:
- Monthly Payment for Incentive B: [tex]\(\frac{46,000}{36}\)[/tex]

### Calculation Results
By performing the necessary detailed calculations using the given values:

- Loan Amount for Incentive A: \[tex]$40,000 - Loan Amount for Incentive B: \$[/tex]46,000
- Monthly Payment for Incentive A: \[tex]$1230.52 (rounded to nearest cent) - Monthly Payment for Incentive B: \$[/tex]1277.78 (rounded to nearest cent)

Finally, the difference in monthly payments is:
[tex]\[ \$1277.78 - \$1230.52 = \$47.26 \][/tex]

### Conclusion:
The difference in monthly payments between the two offers is \[tex]$47.26. Incentive A, which results in a lower monthly payment of \$[/tex]1230.52 compared to Incentive B's monthly payment of \$1277.78, is the better deal.
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