Linda owes $3,861 on a credit card with a 22.3% interest rate compounded monthly. What is the monthly payment she should make in order to pay off this debt in 2 years, assuming she does not charge any more purchases with the card?
I don't need an answer, I just want to know how to do the problem.

Question

Linda owes $3,861 on a credit card with a 22.3% interest rate compounded monthly. What is the monthly payment she should make in order to pay off this debt in 2 years, assuming she does not charge any more purchases with the card?
I don't need an answer, I just want to know how to do the problem.

anonymous · Answer

take  and multiply by .223 then divide that total by 24 months... there is the answer.

anonymous · Answer

*take $3861 and mulyiply

tkhunny · Answer

Set up some rules:

i = 0.223 -- This is the annual interest rate.
j = 0.223/12 = 0.0185833... -- This is the monthly interest rate.
r = 1+j = 1.0185833... -- This is the monthly interest accumulation factor.

Okay, that's all we need.  Now, let's build the balance in the account.

At the beginning, we have 3861.  This is given in the problem statement.

One month later, we have 3861*r - PMT.  This is the interest accumulation for the one month, less the payment that we don't know.

Two months after we started, we have (3861*r - PMT)*r - PMT.  This is the interest accumulation for the next month, less the second equal payment.  See how that works?

Now, we have a little algabra.  The mothod demonstarted above will become very tedious very quickly.  We'll need to realize how it will turn out and simplify it!  At the very end, the day we pay off the loan, we'll have this:

$3861*r^{24} - PMT(r^{23} + r^{22} + ... + r^{0}) = 0$

This represents a valuation on the day things are paid off.  We can also provide a valuation on the day we start.

If we define v = 1/r -- a monthly discount factor, 

$3861 = PMT(v + v^{2} + ... + v^{24})$

Either way works fine.  The remining challenge is adding up those Geometric Series.  Can you do that?

anonymous · Answer

@jondominguez Do you know which formula should you apply with?

tkhunny · Answer

Of course, if you know how to add those geometric series, you write your own formulas.

$v + v^{2} + ... + v^{n} = \dfrac{v - v^{n+1}}{1-v} = \dfrac{1-v^{n}}{j}$

anonymous · Answer

Geometric series is out of the question. That makes the problem a whole lot more complicated but it's fine, I found the equation, thanks for helping!

tkhunny · Answer

No, it IS a Geometric Series.  It is exactly the defintion of a Geometric Series and everything you ever studied about Geometric Series is applicable to this problem.

anonymous · Answer

M=B (i(1+i)^nt/ (1+i)^nt -1  \]