Financial institutions often offer lower auto loan rates for new cars than used ones. A local credit union advertises new car loans at 2.79% APR and used car loans at 3.29% both for up to 72 months. Tyresa wants to buy a car but doesn’t want to spend more than $350 a month for a maximum of four years. What is the maximum loan amount she can take out for a new car and a used car using the advertised rates? Use the formula A = P[(1+ ) −1] rn nt rn (1+ ) , where P is the monthly payment, r is the annual interest rate, n is the number times interest is compounded in one year, and t is the number of years. To receive full credit, be sure to show all steps and calculations used, included where you rounded.
First, let's calculate the maximum loan amount Tyresa can take out for a new car under the advertised rate of 2.79% APR.
Using the formula A = P[(1+ r/n)^(nt) -1], we can plug in the values:
P = $350 (maximum monthly payment)
r = 0.0279 (annual interest rate of 2.79%)
n = 12 (interest is compounded monthly)
t = 4 (4 years)
A = P[(1 + r/n)^(nt) - 1]
A = $350[(1 + 0.0279/12)^(12*4) - 1]
A = $350[(1 + 0.002325)^(48) - 1]
A = $350[(1.002325)^48 - 1]
A = $350[1.12327 - 1]
A = $350[0.12327]
A = $43.1425
Therefore, the maximum loan amount Tyresa can take out for a new car is $43.14.
Now, let's calculate the maximum loan amount for a used car under the advertised rate of 3.29% APR using the same formula.
P = $350 (maximum monthly payment)
r = 0.0329 (annual interest rate of 3.29%)
n = 12 (interest is compounded monthly)
t = 4 (4 years)
A = P[(1 + r/n)^(nt) - 1]
A = $350[(1 + 0.0329/12)^(12*4) - 1]
A = $350[(1 + 0.0027425)^(48) - 1]
A = $350[(1.0027425)^48 - 1]
A = $350[1.14793 - 1]
A = $350[0.14793]
A = $51.7485
Therefore, the maximum loan amount Tyresa can take out for a used car is $51.75.