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+rn)nt−1]rn(1+rn)nt
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. Show all of your steps.
To solve for the maximum loan amount, we can rearrange the formula to solve for principal (P).
For a new car loan:
P = A / [(1+rn)^(nt) - 1] / rn
P = 350 / [(1 + 0.0279/12)^(12 * 4) - 1] / (0.0279/12)
Simplifying the expression:
P = 350 / [(1.002325)^48 - 1] / 0.002325
Calculating the numerator:
Numerator = [(1.002325)^48 - 1]
Numerator ≈ 0.1255
Calculating the denominator:
Denominator = 0.002325
Denominator ≈ 0.002325
Simplifying the expression:
P = 350 / 0.1255 / 0.002325
P = 350 / 0.0539
Calculating the principal:
P ≈ $6,487.75
Therefore, the maximum loan amount Tyresa can take out for a new car is approximately $6,487.75.
For a used car loan:
P = A / [(1+rn)^(nt) - 1] / rn
P = 350 / [(1 + 0.0329/12)^(12 * 4) - 1] / (0.0329/12)
Simplifying the expression:
P = 350 / [(1.002741)^48 - 1] / 0.002742
Calculating the numerator:
Numerator = [(1.002741)^48 - 1]
Numerator ≈ 0.1311
Calculating the denominator:
Denominator = 0.002742
Denominator ≈ 0.002742
Simplifying the expression:
P = 350 / 0.1311 / 0.002742
P = 350 / 0.0478
Calculating the principal:
P ≈ $7,311.32
Therefore, the maximum loan amount Tyresa can take out for a used car is approximately $7,311.32.