Questions: Suppose that you are thinking about buying a car and have narrowed down your choices to two options. The new-car option: The new car cost 29,000 and can be financed with a three-year loan at 7.84%. The used-car option: A three-year old model of the same car costs 14,000 and can be financed with a four-year loan at 5.43%. The difference in monthly payments between financing the new car and financing the used car is . (round to the nearest cent as needed.)

Suppose that you are thinking about buying a car and have narrowed down your choices to two options.

The new-car option: The new car cost 29,000 and can be financed with a three-year loan at 7.84%.
The used-car option: A three-year old model of the same car costs 14,000 and can be financed with a four-year loan at 5.43%.
The difference in monthly payments between financing the new car and financing the used car is . (round to the nearest cent as needed.)
Transcript text: Suppose that you are thinking about buying a car and have narrowed down your choices to two options. The new-car option: The new car cost $\$ 29,000$ and can be financed with a three-year loan at $7.84 \%$. The used - car option: A three-year old model of the same car costs \$14,000 and can be financed with a four-year loan at $5.43 \%$. The difference in monthly payments between financing the new car and financing the used car is $\$ \$$. (round to the nearest cent as needed.)
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Solution

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Solution Steps

To find the difference in monthly payments between financing the new car and the used car, we need to calculate the monthly payment for each car using the loan amount, interest rate, and loan term. We can use the formula for monthly payments on an installment loan:

M=Pr(1+r)n(1+r)n1 M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}

where:

  • M M is the monthly payment
  • P P is the loan principal (the amount of the loan)
  • r r is the monthly interest rate (annual rate divided by 12)
  • n n is the number of payments (loan term in months)

After calculating the monthly payments for both the new car and the used car, we subtract the used car's monthly payment from the new car's monthly payment to find the difference.

Step 1: Calculate Monthly Payment for New Car

The new car costs P=29000 P = 29000 and has an annual interest rate of r=0.0784 r = 0.0784 over a term of n=3 n = 3 years. The monthly payment M M can be calculated using the formula:

M=Prm(1+rm)N(1+rm)N1 M = \frac{P \cdot r_m \cdot (1 + r_m)^N}{(1 + r_m)^N - 1}

where rm=r12=0.078412 r_m = \frac{r}{12} = \frac{0.0784}{12} and N=n12=312=36 N = n \cdot 12 = 3 \cdot 12 = 36 .

Substituting the values, we find:

Mnew=290000.078412(1+0.078412)36(1+0.078412)361906.6156 M_{\text{new}} = \frac{29000 \cdot \frac{0.0784}{12} \cdot (1 + \frac{0.0784}{12})^{36}}{(1 + \frac{0.0784}{12})^{36} - 1} \approx 906.6156

Step 2: Calculate Monthly Payment for Used Car

The used car costs P=14000 P = 14000 with an annual interest rate of r=0.0543 r = 0.0543 over a term of n=4 n = 4 years. Using the same formula for monthly payment:

Mused=14000rm(1+rm)N(1+rm)N1 M_{\text{used}} = \frac{14000 \cdot r_m \cdot (1 + r_m)^N}{(1 + r_m)^N - 1}

where rm=0.054312 r_m = \frac{0.0543}{12} and N=412=48 N = 4 \cdot 12 = 48 .

Substituting the values, we find:

Mused=140000.054312(1+0.054312)48(1+0.054312)481325.1442 M_{\text{used}} = \frac{14000 \cdot \frac{0.0543}{12} \cdot (1 + \frac{0.0543}{12})^{48}}{(1 + \frac{0.0543}{12})^{48} - 1} \approx 325.1442

Step 3: Calculate the Difference in Monthly Payments

The difference in monthly payments between the new car and the used car is given by:

Difference=MnewMused906.6156325.1442=581.4714 \text{Difference} = M_{\text{new}} - M_{\text{used}} \approx 906.6156 - 325.1442 = 581.4714

Rounding to the nearest cent, we have:

Difference581.47 \text{Difference} \approx 581.47

Final Answer

The difference in monthly payments between financing the new car and financing the used car is \\(\boxed{581.47}\\).

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