Question

Joey realizes that he has charged too much on his credit card and has racked up $5,900 in debt. If he can pay $150 each month and the card charges 12 percent APR (compounded monthly), how long will it take him to pay off the debt? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

Answer 1

Solution:
	It will takes 50.23 months to pay off the debt
Working Notes:
	The today debt value $5,900 is the present value of annuity of $150 payment each month.
	present value of annuity = Px[ 1-1 /(1 + i)^n)]/ i
	P=monthly payment =$150
	i= interest rate per period = 12%/12
	n= no. Of period = ??
	PV of annuity= Debt $5,900
	present value of annuity = Px[ 1-1 /(1 + i)^n)]/ i
	5900 = 150 x (1-1/(1+(12%/12))^n)/(12%/12)
	5900 = 150 x (1-1/(1+ 0.01)^n)/0.01
	(5900/150) x 0.01 = (1-1/(1+ 0.01)^n)
	0.393333333 = 1-1/(1+ 0.01)^n
	1/(1+ 0.01)^n = 1- 0.393333333
	1/0.606666667 =(1+ 0.01)^n
	1.648351647 = 1.01^n
	Taking log on both side
	log (1.648351647) = Log(1.01)^n
	using relation loga^b = b x Log a
	log (1.648351647) = Log(1.01)^n
	log (1.648351647) =n x Log(1.01)
	n= log(1.648351647)/log(1.01)
	n= 50.22705216 months
	n=50.23 months
Now By Excel method
	=NPER(rate, pmt,pv,fv)
	P=monthly payment =pmt = -$150
	negative as its cash out flows
	i= interest rate per period = 12%/12=rate =1%
	n= no. Of period = nper =??
	face value = 0 = fv
	end period payment is zero as we pay only monthly payment no balloon payment at end of the period.
	PV of annuity= Debt $5,900=PV=5900
	=NPER(rate, pmt,pv,fv)
	=NPER(1%,-150,5900,0)
	50.22705224
	50.23
	no of months= nper = 50.23 months
Please feel free to ask if anything about above solution in comment section of the question.