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.)

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:

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

where:

• \( M \) is the monthly payment
• \( P \) is the loan principal (the amount of the loan)
• \( r \) is the monthly interest rate (annual rate divided by 12)
• \( n \) 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.

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

\[ M = \frac{P \cdot r_m \cdot (1 + r_m)^N}{(1 + r_m)^N - 1} \]

where \( r_m = \frac{r}{12} = \frac{0.0784}{12} \) and \( N = n \cdot 12 = 3 \cdot 12 = 36 \).

Substituting the values, we find:

\[ M_{\text{new}} = \frac{29000 \cdot \frac{0.0784}{12} \cdot (1 + \frac{0.0784}{12})^{36}}{(1 + \frac{0.0784}{12})^{36} - 1} \approx 906.6156 \]

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

\[ M_{\text{used}} = \frac{14000 \cdot r_m \cdot (1 + r_m)^N}{(1 + r_m)^N - 1} \]

where \( r_m = \frac{0.0543}{12} \) and \( N = 4 \cdot 12 = 48 \).

Substituting the values, we find:

\[ M_{\text{used}} = \frac{14000 \cdot \frac{0.0543}{12} \cdot (1 + \frac{0.0543}{12})^{48}}{(1 + \frac{0.0543}{12})^{48} - 1} \approx 325.1442 \]

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

\[ \text{Difference} = M_{\text{new}} - M_{\text{used}} \approx 906.6156 - 325.1442 = 581.4714 \]

Rounding to the nearest cent, we have:

\[ \text{Difference} \approx 581.47 \]

Final Answer