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Mr.Debt has 3 credit cards. First card have \$5,000 balance with the rate of 10% compounded monthly. Second card have \$2,000 balance with the rate of 14% compounded monthly. Third card have \$4,000 balance with the rate of 8% compounded monthly. All card have minimal payment of 1% + each month interest . Mr.Debt want to set a fix amount of payment per month. How much will Mr.Debt will have to pay as little as possible and still payoff his debt in 24 months.

Please Help.

I found the formula to do a single credit card which is http://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.321982.html but does not seem to work with multiple credit card.

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Even though you've not mentioned this is homework, I feel it might be. So if I am right and since you're new, you may want to read how to post homework questions here meta.math.stackexchange.com/questions/1803/… –  haunted85 Dec 7 '12 at 16:20
It is not really homework but ill put the tag on since i do need explanation. What I am trying to do is create a program that could do the math above. But I do need to know how to calculate it first which i don't. Thank for let me know about homework question. –  user1883251 Dec 7 '12 at 16:31
Where did you get this question? It's pretty complicated. I mean, first you're going to pay off the second card because it has the highest interest rate. Then, you will do the first, and finally the third. But, all along, you're going to be making the minimum payments to the other cards. Since the amount of each payment is unknown, the amount of time it's going to take you to pay off the second card is an unknown. And, that fact makes it harder to find the amount of each payment. There's also the time it's going to take to finish the first card. That is, there are many variables here. –  Graphth Dec 7 '12 at 16:36
This question is created by me. As I comment above that i need to program some thing to solve something similar to the question so. and the minimum payment is 1% of the balance of each month + the interest it self. I do not know how credit card work exactly but this is what i think , if you have 12% apr / 12 month you will have 1% monthly interest. and minimum payment is 1% + 1% interest so I think it would be 2% of the balance of each month as minimum payment. I might be wrong but this is what i understand. If you know correct way Please let me know. –  user1883251 Dec 7 '12 at 16:39
Essentially, you're making three different payment amounts each month, and each one affects the others. And, how long you will make two of the payments is unknown. In the real world, what you would do is try to get a new credit card with a 0% interest rate and transfer your balances to that one. :) –  Graphth Dec 7 '12 at 16:41
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1 Answer

To actually solve your problem, you would start out with an unknown amount of $X$ payment per month. This is the variable you care about. In fact, you would program a function where this is your input. You would then program an iterative process that pays the minimum on the two cards with the lower rates and the rest of $X$ on the one with the highest balance, until that one is paid off. Once that one is paid off, you start paying the minimum on the one with 8\% interest and pay all the rest to the one with 10\%. And, you would make payments for 24 months. And, once the the second is paid off, all remaining payments go to the one remaining card.

Now that you have this function programmed, you run it with different values of $X$ until your remaining value is \$0. Use some basic knowledge, such as, if one value of $X$, say $X_1$, gives you a positive balance at the end, and another, say $X_2$, gives you a negative balance at the end, then your answer is somewhere in between.

By the way, in reality, the formula you found is wrong. A 12% annual interest rate does NOT translate into 1% per month. But, maybe that is how they teach things when you first see this stuff.

$$A(t) = A_0 \left( 1 + \frac{r}{n} \right)^{nt}$$

is the formula you can use to find the amount, $A(t)$, at time $t$ if your initial balance is $A_0$, your interest rate is $r$ (given as a decimal, so 12% would be 0.12), and $n$ is the number of payments per month. An annual rate of 12% per year means that if you start with \$100, after 1 year, you will have 12%. A monthly rate of 1% means after 1 month, you would have $100 (1 + .01) = 101$. And, in general, after $k$ months, you'd have $100 (1.01)^k$. So, after 12 months, you'd actually have \$112.68. It's somewhat close, but it's not the same.

To take an annual rate, $r$, and translate it into the appropriate monthly rate, you would want to solve $$1 + r = \left( 1 + \frac{i}{12} \right)^{12}$$ for $i/12$, which would give you $$\frac{i}{12} = (1+r)^{1/12} - 1.$$

So, for instance, if your annual rate is 14%, your monthly rate would be $$\frac{i}{12} = (1.14)^{1/12} - 1 = 0.010978852.$$ This is not the same as 0.14/12 = 0.011666666666.

On the other hand, if you said that your interest rate is 14% compounded monthly, then your monthly rate is in fact 0.14 / 12. It's a matter of knowing the terminology and it is important in reality. Just to be clear, if you say your rate is an annual rate of 14%, then the above calculations are necessary. If you say the rate is 14% compounded monthly, then the monthly rate is 0.14/12.

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You could also use Integer Linear Programming. But I agree in that I cannot see any purely analytic solution to this.. –  Fab Dec 7 '12 at 16:49
wasn't 12% /12 = 1% monthly is how compounded interest work ? I thought that how it work. I guess i have the wrong idea the whole time then. Thank you for help. –  user1883251 Dec 7 '12 at 17:01
Thank for let me know the different ill fix my question to compounded. –  user1883251 Dec 7 '12 at 17:04
@user1883251 Yea, it's confusing at first. If you say it is an annual rate, then it is the rate for one year. If you say it is compounded monthly, then that's different. So, if a bank tells you your loan is at 7%, it's important to know if they mean annual rate or compounded monthly or compounded weekly or whatever. –  Graphth Dec 7 '12 at 17:16
Sorry about that. I don't know much about this stuff. And Thank you. –  user1883251 Dec 7 '12 at 17:18
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