Suppose you want to buy a car. The dealer offers a financing package consisting of a 3.6% APR compounded monthly for a term of four years. Suppose that you want your monthly payments to be at most $500. What is the maximum amount that you should finance?
To find the maximum amount that you should finance, we need to use the formula for calculating the monthly payment on a loan:
M = P [ r(1+r)^n ] / [ (1+r)^n - 1 ]
Where:
M = Monthly payment
P = Principal amount (amount being financed)
r = Monthly interest rate
n = Number of monthly payments
In this case:
M = $500
r = 3.6% / 12 = 0.003
n = 4 years * 12 months = 48
Now, we can plug these values into the formula and solve for P:
$500 = P [ 0.003(1.003)^48 ] / [ (1.003)^48 - 1 ]
Simplifying the equation:
$500 = P [ 0.003 x 1.20492 ] / [ 1.20492 - 1 ]
$500 = P [ 0.00361476 ] / [ 0.20492 ]
$500 = P x 0.017647
Dividing both sides by 0.017647:
P = $500 / 0.017647
P ≈ $28363.10
Therefore, the maximum amount you should finance is approximately $28,363.10.