Question

​(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?

Answer 1

Answer:

41.49 approx 42 months

Explanation:

To calculate the number of months, we use the formula for loan

p = r(pv) / 1 - (1+r)-n

make n subject of the formula

p ( 1 - ( 1+r) ^-n) = r(pv)

p - p (1+r)^-n = r(pv)

p (1+r)^-n = p-r(pv)

(1+r)^-n = (p-r(pv)) / p

( 1+r)^n = p / (p-r(pv))

n In( 1+r) = In (p / (p-r(pv))

n = In ( p/ ( p - r(pv)) / In ( 1 +r)

n is the number of months, p is the payment per months

pv is the present value of 5000

substitute the values given into the equation

n = (In ( 150 / (150 - ( 0.129 / 12 × 5000)) / ( In ( 1 + ( 0.129 / 12) = 41.49 approx 42 months

Answer 2

Answer:

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

$n = log_{1.01075}1.56$

n = 41.59  approximately 42 months.

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