To solve this problem, we need to calculate the monthly payments for both the new car and the used car using the provided loan formula:
[tex]\[
PMT = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\][/tex]
Here:
- [tex]\(P\)[/tex] is the principal loan amount (the cost of the car).
- [tex]\(r\)[/tex] is the annual interest rate.
- [tex]\(n\)[/tex] is the number of payments per year.
- [tex]\(t\)[/tex] is the loan term in years.
For the new car:
- Principal, [tex]\(P_{\text{new}}\)[/tex] = \[tex]$ 28,000
- Annual interest rate, \(r_{\text{new}}\) = 0.0522
- Loan term, \(t_{\text{new}}\) = 5 years
- Number of payments per year, \(n = 12\)
Plug these values into the formula:
\[
PMT_{\text{new}} = \frac{28000 \left(\frac{0.0522}{12}\right)}{1 - \left(1 + \frac{0.0522}{12}\right)^{-12 \times 5}}
\]
For the used car:
- Principal, \(P_{\text{used}}\) = \$[/tex] 14,000
- Annual interest rate, [tex]\(r_{\text{used}}\)[/tex] = 0.0786
- Loan term, [tex]\(t_{\text{used}}\)[/tex] = 4 years
- Number of payments per year, [tex]\(n = 12\)[/tex]
Plug these values into the formula:
[tex]\[
PMT_{\text{used}} = \frac{14000 \left(\frac{0.0786}{12}\right)}{1 - \left(1 + \frac{0.0786}{12}\right)^{-12 \times 4}}
\][/tex]
After calculating these monthly payments, we can find the difference between the monthly payment for the new car and the used car.
Performing the calculations:
1. For the new car, the monthly payment (PMT_new) is approximately [tex]$531.22.
2. For the used car, the monthly payment (PMT_used) is approximately $[/tex]340.86.
To find the difference in monthly payments:
[tex]\[
\text{Difference} = PMT_{\text{new}} - PMT_{\text{used}} = 531.22 - 340.86 = 190.36
\][/tex]
So, the difference in monthly payments between financing the new car and financing the used car is [tex]\(\$ 190.36\)[/tex].
