Like many college students, Bernadette applied for and got a credit card that has an annual percentage rate (APR) of 15%. The first thing she did was buy a new stereo system for $400. At the end of the month, her credit card statement said she only needed to make a minimum monthly payment of $15. Assume Bernadette makes her payment when she sees her statement at the end of each month. If Bernadette doesn't charge anything else and only makes the minimum monthly payments, approximately how many months would it take her to completely pay off the stereo system? Assume that the credit card company compounds interest at the end of each month.

It would take Bernadette approximately 33 months to completely pay off the stereo system.

Explanation:

Since it is assumed that Bernadette makes her payment when she sees her statement at the end of each month,  we use the formula for calculating the present value (PV) of an ordinary annuity to determine the number of months as follows:

PV = M * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)

Where;

PV = Present value of the new stereo system = $400

M = minimum monthly payment = $15

r = monthly interest rate = 15% / 12 = 0.15 / 12 = 0.0125

n = number of months = n

Substitute the values into equation (1) and solve for n as follows:

400 = 15 * ((1 - (1 / (1 + 0.0125))^n) / 0.0125)

400 / 15 = (1 - (1 / 1.0125)^n) / 0.0125

26.6666666666667 * 0.0125 = 1 - (1 / 1.0125)^n

0.333333333333334 = 1 - 0.987654320987654^n

0.987654320987654^n = 1 - 0.333333333333334

0.987654320987654^n = 0.666666666666666

Loglinearizing both sides, we have:

n * log0.987654320987654 = log0.666666666666666

n * (-0.00539503188670629) = -0.176091259055682

n = -0.176091259055682 / -0.00539503188670629

n = 32.64, or 33 approximately.

Therefore, it would take Bernadette approximately 33 months to completely pay off the stereo system.

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