(Annuity payments) Calvin Johnson has a 5000 debt balance on his Visa card that charges 12.9 percent APR compounded monthly. In 2009 , Calvin's minimum monthly payment is 3 percent of his debt balance, which is $150. How many months (round up) will it take Calvin Johnson to pay off his credit card if he pays the current minimum payment of $150 at the end of each month?

42 months

Explanation:

To calculate the number months, we use ordinary annuity formula since that payment is expected to be made at the end of each month. The ordinary annuity formula is as follows:

PV = P × [{1 - [1 ÷ (1 + r)]^n} ÷ r]  .............................. (1)

Where ,

PV = Present value of an annuity  payment = $5,000

P = monthly payment  = $150

r = interest rate  = 12.9% annually = 0.129 annually = (0.129 ÷ 12) monthly =  0.01075  monthly

n = number of months = ?

Substituting the values into equation (1), we have:

5000 = 150 × [{1 - [1 ÷ (1 + 0.01075)]^n} ÷ 0.01075]  

5000 ÷ 150 = {1 - [1 ÷ (1.01075)]^n} ÷ 0.01075

33.33 × 0.01075 = 1 - [1 ÷ (1.01075)]^n

1 - 0.36 = [1 ÷ (1.01075)]^n

0.64 = 1^n ÷ (1.01075)^n

Since 1^n = 1, we have:

0.64 = 1 ÷ (1.01075)^n

Rearranging, we have:

(1.01075)^n = 1 ÷ 0.64

(1.01075)^n = 1.56

By converting the exponential function to a logarithm function, we have:

n = log_{1.01075}1.56

n = 41.59  approximately 42 months.

Therefore, it will take Calvin Johnson 42 months to pay off his credit card.

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