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.
