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=(1+r)n−1P⋅r⋅(1+r)n
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=(1+rm)N−1P⋅rm⋅(1+rm)N
where rm=12r=120.0784 and N=n⋅12=3⋅12=36.
Substituting the values, we find:
Mnew=(1+120.0784)36−129000⋅120.0784⋅(1+120.0784)36≈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:
Mused=(1+rm)N−114000⋅rm⋅(1+rm)N
where rm=120.0543 and N=4⋅12=48.
Substituting the values, we find:
Mused=(1+120.0543)48−114000⋅120.0543⋅(1+120.0543)48≈325.1442
The difference in monthly payments between the new car and the used car is given by:
Difference=Mnew−Mused≈906.6156−325.1442=581.4714
Rounding to the nearest cent, we have:
Difference≈581.47
The difference in monthly payments between financing the new car and financing the used car is \\(\boxed{581.47}\\).