Suppose that you decide to buy a car for $58 comma 000
,
including taxes and license fees. You saved $ 11 comma 000
for a down payment. The dealer is offering you a choice between two incentives.
Incentive A is $6000
off the price of the car, followed by a four
-year
loan at 7.08
%.
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 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 EndEndFraction
.
For incentive A, the loan amount would be $58,000 - $6,000 (rebate) - $11,000 (down payment) = $41,000. Using the loan payment formula P[r/n]/[1-(1+r/n)^(-nt)], where P is the loan amount, r is the annual interest rate in decimal form (0.0708), n is the number of payments per year (12), and t is the number of years (4), the monthly payment would be calculated as $41,000[0.0708/12]/[1-(1+0.0708/12)^(-12*4)] = $1,000.32.
For incentive B, the loan amount would simply be $58,000 - $11,000 (down payment) = $47,000, and because there's no interest, the monthly payment would simply be $47,000 / (4 * 12) = $979.17.
So the difference in monthly payments between the two offers would be $1,000.32 - $979.17 = $21.15.
The incentive with the lower monthly payment is incentive B (the no-interest loan), so that would be the better deal.
In conclusion, the difference in monthly payments between the two offers is $21.15, and Incentive B is the better deal.
To find the difference in monthly payments between the two offers, we need to calculate the monthly payments for each incentive option.
For Incentive A:
1. Calculate the total cost of the car after the down payment: $58,000 - $11,000 = $47,000.
2. Apply the $6,000 discount to the price of the car: $47,000 - $6,000 = $41,000.
3. Use the formula PMT = P * (r/n) / (1 - (1 + r/n)^(-nt), where P is the principal amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
- P = $41,000
- r = 7.08% = 0.0708 (converted to decimal)
- n = 12 (since there are 12 months in a year)
- t = 4 years
Applying these values to the formula, we get PMT = 41000 * (0.0708/12) / (1 - (1 + 0.0708/12)^(-4*12)).
After evaluating this expression, we find that the monthly payment for Incentive A is approximately $925.34.
For Incentive B:
Since there is no interest, the monthly payment for Incentive B will be 0.
Now, let's calculate the difference in monthly payments:
Difference = Payment for Incentive A - Payment for Incentive B
Difference = $925.34 - $0
Difference = $925.34
The difference in monthly payments between the two offers is approximately $925.34.
To determine which incentive is the better deal, we need to consider other factors such as the total cost of the car and the value of the incentives. In this case, the better deal would depend on individual preferences and financial goals. Incentive A provides a $6,000 cash rebate but has monthly payments, while Incentive B offers no cash rebate but has no monthly payments.