Your Christmas ski vacation was great, but it unfortunately ran a bit over budge
ID: 2775872 • Letter: Y
Question
Your Christmas ski vacation was great, but it unfortunately ran a bit over budget. All is not lost, because you just received an offer in the mail to transfer your $12,600 balance from your current credit card, which charges an annual rate of 20.4 percent, to a new credit card charging a rate of 11 percent.
How much faster could you pay the loan off by making your planned monthly payments of $255 with the new card? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)
Number of months? _____
What if there was a 2 percent fee charged on any balances transferred? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)
Number of months? _____
How much faster could you pay the loan off by making your planned monthly payments of $255 with the new card? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)
Number of months? _____
What if there was a 2 percent fee charged on any balances transferred? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)
Number of months? _____
Explanation / Answer
By using the new card and an EMI of $ 255 the loan could be paid in 66 months which is 43 months faster than the existing card.
By using the new card with a 2% charge Balances transferred and an EMI of $ 255 the loan could be paid in 68 months which is 41 months faster than the existing card.
working
Outstanding Balance on card = $ 12600
Annual Interest rate with existing card = 20.4% or 1.7% per month
Planned Monthly instalment = $ 255
Let n1 be the number of months taken to pay off the loan with EMI of $ 255. Then
$ 255 = $ 12600 * [(0.017 * (1+0.017)^n1)/(1+0.017)^n1 – 1)]
$ 255 ((1.017)^n1 – 1) = $ 12600 * [(0.017 * (1.017)^n1)
$ 255. 1.017^n1 - $ 255 = $ 214.2 * 1.017^n1 ==> $ 255. 1.017^n1 - 214.2 * 1.017^n1 = $ 255
1.017^n1 * ($255 - $ 214.2) = $ 255 ==> 1.017^n1 * $ 40.8 = $ 255
1.017^n1 = $255/$40.8 ==> 1.017^n1 = 6.25
Taking logarithms on both sides
log 1.017^n1 = log 6.25 ==> n1 * log 1.017 = log 6.25 ==> n1 = log 6.25/log 1.017
n1 = 0.79588/0.00732 ==> n1 = 108.72 or 109 months
on transfer to new card interest rate is 11% per annum or 0.9167% per month
Let n2 be the number of months taken to pay off the card loan with EMI of $ 255. Then
$ 255 = $ 12600 * [(0.009167 * (1+0.009167)^n2)/(1+0.009167)^n2 – 1)]
$ 255 ((1.009167)^n2 – 1) = $ 12600 * [(0.009167 * (1.009167)^n2)
$ 255. 1.009167^n2 - $ 255 = $ 115.50 * 1.009167^n2
$ 255. 1.009167^n2 - $ 115.50 * 1.009167^n2 = $ 255
1.009167^n2 * ($255 - $ 115.50) = $ 255 ==> 1.009167^n2* $ 139.50 = $ 255
1.009167^n2 = $255/$139.50 ==> 1.009167^n2 = 1.827957
Taking logarithms on both sides
log 1.009167^n2 = log 1.827957 ==> n2 * log 1.009167 = log 1.827957
n2 = log 1.827957/log 1.009167 ==> n2 = 0.261966/0.003963 = 66.10 or 66 months
By using the new card the loan could be paid in n1-n2 = 109 – 66 = 43 month faster
If there a 2% charge on balance transfer to new card, then the balance on new card after transfer would be $12600 * 1.02 = $ 12852
No of months take to pay the amount with an EMI of $ 255 would be
Let n2 be the number of months taken to pay off the card loan with EMI of $ 255. Then
$ 255 = $ 12852 * [(0.009167 * (1+0.009167)^n2)/(1+0.009167)^n2 – 1)]
$ 255 ((1.009167)^n2 – 1) = $ 12852 * [(0.009167 * (1.009167)^n2)
$ 255. 1.009167^n2 - $ 255 = $ 117.81 * 1.009167^n2
$ 255. 1.009167^n2 - $ 117.81 * 1.009167^n2 = $ 255
1.009167^n2 * ($255 - $ 117.81) = $ 255 ==> 1.009167^n2* $ 137.19 = $ 255
1.009167^n2 = $255/$137.19 ==> 1.009167^n2 = 1.858736
Taking logarithms on both sides
log 1.009167^n2 = log 1.858736 ==> n2 * log 1.009167 = log 1.858736
n2 = log 1.858736/log 1.009167 ==> n2 = 0.2692177/0.003963 = 67.93 or 68 months
By using the new card with a 2% charge on balance transfer the loan could be paid in n1-n2 = 109 – 68 = 41 month faster
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