Credit Card DebtDate: 05/09/97 at 23:02:43 From: James Frost Subject: Financial Formula or Function I am retired and long out of school. The only devices we had for math were a slide rule, pencils and a lot of erasers. Assume a credit card debt of $1000 at an APR of .155% which is paid back at $50 per month. How long will it take to pay off the amount and what will be the total amount paid to erase the debt? How do the credit card companies figure this out? Date: 05/10/97 at 07:20:14 From: Doctor Mitteldorf Subject: Re: Financial Formula or Function Dear James, If you're comfortable with algebra and geometric series, then problems like this are much easier than if you're just working with multiplication and subtraction. Let the monthly interest rate be r and the principal be p and the monthly payment be x. Then each month p becomes p*(1+r) - x. If this happens 2 times, then you have: p*(1+r)^2 - x*(1+r) - x After n months, you have: p*(1+r)^n - x*[ (1+r)^(n-1) + (1+r)^(n-2) + (1+r)^(n-3) .. + (1+r)^0 ] The first term is just the nth power of (1+r). The rest of the terms form the sum of a geometric series in (1+r). Way back when you were in high school, you learned a trick for finding the sum of a geometric series. If you multiply every term by (1+r), you get back the same series you had, except the last term is missing and there's a new, larger term in the front. With a little algebra, this insight is turned into a formula for the sum of a geometric series. You may have a good time working out the details yourself. The result is: p = x*(1 - (1+r)^-n)/r In words: take the negative nth power of (1+r). Subtract from 1, divide by r. This gives the ratio of p, the initial principal, to x, the monthly payment. This formula will let you solve for x, the monthly payment that will pay off your principal p in n months. You can also solve backwards for p given x. If you're comfortable with logarithms, you can also solve for n, the number of months it will take you to pay off the loan. For r, the monthly rate, you should just use 1/12 of the APR. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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