Problem 5-37 Paying off credit cards Simon recently received a credit card with
ID: 2777821 • Letter: P
Question
Problem 5-37
Paying off credit cards Simon recently received a credit card with an 13% nominal interest rate. With the card, he purchased an Apple iPhone 5 for $400. The minimum payment on the card is only $10 per month.
a) If Simon makes the minimum monthly payment and makes no other charges, how many months will it be before he pays off the card? Round your answer to the nearest whole. month(s)
b) If Simon makes monthly payments of $30, how many months will it be before he pays off the debt? Round your answer to the nearest whole. month(s)
c) How much more in total payments will Simon make under the $10-a-month plan than under the $30-a-month plan. Make sure you use 3 decimal places for N. Round your answer to the nearest cent.
Explanation / Answer
a)Let T be the months will it be before he pays off the card,minimum monthly payment=MP=10,Initial balance or amount =MB0=400,nominal interest rate=13%=>monthly interest rate=r=13%/12=.13/12
We know that a monthly payment MP required for intial amount MB0 taken at interest r for time T,Monthly payment=MP=MB0*r/(1-1/(1+r)T)
=> (1-1/(1+r)T) = MB0*r/MP
= > 1/(1+r)T=1-(MB0*r/MP)
= > (1+r)T=1/(1-(MB0*r/MP))
take ln on both sides => T=ln(1/(1-(MB0*r/MP)))/ln(1+r)
Put above values in the above formula =>T=ln(1/(1-(400*(.13/12)/10)))/ln(1+(.13/12))
=>T=ln(1/(1-(.4333)))/ln(1.010833)
=>T=ln(1.764)/ln(1.010833)=.5674/.0107747=52.660=52.7~53
Thus 53 months would be required before he pays off the card.
b)If MP=30,T=ln(1/(1-(MB0*r/MP)))/ln(1+r)=>T=ln(1/(1-(400*(.13/12)/30)))/ln(1+(.13/12))
=>T=ln(1/(1-.1444))/ln(1+(.13/12))
=>T=ln(1.16877))/ln(1.010833)=.155952/.0107747=14.474=14.5 =~15
Thus 15 months would be required before he pays off the card when monthly payments is of $30.
c) Total payments will Simon make under the $10-a-month plan than under the $30-a-month plan in excess=52.660-14.474=38.186=~38
Thus 38 more payments will Simon make under the $10-a-month plan than under the $30-a-month plan.