Question 1171661: You have a balance of $2600 on your Visa credit card. Assume that you make no more charges on the card and that the card charges 9.6% APR and requires a minimum payment of 3% of the balance. Assume also that you make only the minimum payments.
Find a formula for the balance B after t monthly payments.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the interest charge is 9.6% per year.
payments are usually monthly, with monthly compounding assumed.
the interest 9.6% per per year is divided by 12 to get an interest rate of 9.6/12 = .8% per month.
the formula uses the rate, not the percent.
without any payments, the balance at the end of each month is the balance at the end of the previous month times 1.008.
your minimum payment per month is 3% of the remaining balance.
you would first multiply the remaining balance from the previous month by 1.008 and then multiply that by (1 - .03) = .97.
the net effect is to multiply the remaining balance from the previous month by 1.008 and then multiplying it by .97.
the composite rate is therefore 1.008 * .97 = .97776.
the remaining balance at the end of each month will be .97776 times the remaining balance for the previous month.
at the end of the first month:
2600 * .97776 = 2541.176
at the end of the second month:
2541.176 * .97776 = 2485.638006
at the end of the third month:
2485.638006 * .97776 = 2430.357417
this continues at the end of any month of the loan.
the formula for the balance at the end of any month is:
Bn = 2600 * .9776 ^ n
Bn is the remaining balance for the end of the month indicated.
using this formula, you see that B3 = 2600 * .97773 ^ 3 = 2430.357417.
this is the remaining balance at the end of the third month as indicated earlier.
this proves that the formula can provide you with the balance at the end of any month indicated.
replace n with t and the formula is:
Bt = 2600 * .97776 ^ t
that is the formula for the balance B after t monthly payments.
use that formula to find the balance at the end of 5 monthly payments.
Bt = 2600 * .97776 ^ t becomes B5 = 2600 * .97776 ^ 5 = 2323.457216.
start with end of month 3 balance from above.
at the end of the third month:
2485.638006 * .97776 = 2430.357417
at the end of the fourth month:
2430.357417 * .97776 = 2376.306268
at the end of the fifth month:
2376.306268 * .97776 = 2323.457216.
the formula accurately reflects the remaining balance after t payments.
i'll be available to answer any questions you might have about this.
theo
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