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### Credit Card Debt

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Date: 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?
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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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Associated Topics:
High School Interest

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