UPSB v3
Off-topic / Help me with this math problem?
-
Date: Fri, Apr 16 2010 05:27:07
You've spent $2500 with your new credit card which has an APR of 12.99% compounded monthly. Paying only the minimum payment each month, 1/20 of the outstanding balance, how long will it take to get out of debt and how much money will you have spent doing so? Also, if the balance is $20 or less the whole balance must be paid.
-
Date: Fri, Apr 16 2010 05:39:52
well lets start by making x te amount of money owed.
n shall be the amount of months.
t can be total money payed
so we'll say
20 = 2500 + (((0.1299/12)x) - 0.05x)^n
and then scrap the whole idea coz i dnt know what im doing...... do u need to show working out? i could make a java program to solve it probably but it wouldnt help in a test XD -
Date: Fri, Apr 16 2010 06:13:44QUOTE (Pen Ninja @ Apr 15 2010, 10:39 PM) <{POST_SNAPBACK}>well lets start by making x te amount of money owed.
n shall be the amount of months.
t can be total money payed
so we'll say
20 = 2500 + (((0.1299/12)x) - 0.05x)^n
and then scrap the whole idea coz i dnt know what im doing...... do u need to show working out? i could make a java program to solve it probably but it wouldnt help in a test XD
This isn't testable material or anything. Even advice on how to use a spreadsheet to solve this should be fine. -
Date: Fri, Apr 16 2010 06:20:12
let Ai be amount of money left after i months (yeah, the i in Ai is meant to be subscripted...but w/e)
so:
A0 = $2500 (as initially the amount remaining is $2500 before you make any repayments and before any interest has been added)
to find A1, we know that after 1 month, the original $2500 debt would have increased due to interest. so A1 = A0*(12.99%/12 + 1) - 1st repayment. [derived from using compound interest formula for 1 month, rate is 12.99%/12]. the repayment is 1/20th of the amount you owe before the interest is calculated, so repayment = A0/20.
therefore we have:
A1 = A0*(0.1299/12 + 1) - A0/20 --> = A0*(0.1299/12 + 19/20)
in finding A2, we have to remember that the compound interest and repayment are calculated with A1 instead of A0. so, A2 = A1*(0.1299/12 + 1) - repayment. [using compound interest formular for 1 month again]. you only repay 1/20th of balance owing, so repayment = A1/20 (coz at that point you still owe $A1).
therefore we have:
A2 = A1*(0.1299/12 + 1) - A1/20 --> = A1*(0.1299/12 + 19/20)
so A2 = A0*(0.1299/12 + 19/20)^2 [subbing in value of A1 from previous]
and so on...let n be number such that, immediately after nth payment, An < $20. (as when balance owing is less than $20, you repay the rest of it next month)
An = A0*(0.1299/12 + 19/20)^n = 2500*(0.1299/12 + 19/20)^n. (from observing previous pattern of A1 and A2)
since An < 20, use log rules to solve for n...
2500*(0.1299/12 + 19/20)^n < 20
(0.1299/12 + 19/20)^n < 20/2500
n log (0.1299/12 + 19/20) < log (20/2500).
solving for n, we get: A121 = $19.86
so in 122nd month, you should pay off rest of balance. (see below though, idk)
however, A122 = A121*(0.1299/12 + 19/20) - repayment....note: A121*(0.1299/12 + 19/20) > $20 =__+
(i'm not sure about this, coz technically after 121st month balance is less than $20. So it seems in 122nd month, you'd have to pay rest of it off. But by convention, it is taken that interest is calculated before repayments are made).
ok, so to find how much money you spent, we need to find sum of repayments.
we know 1st repayment = A0/20, 2nd repayment = A1/20, 3rd repayment = A2/20 etc....
from previous, we have found way to express Ai in terms of A0. then use sum of geometric sequence Sn = a(r^n - 1)/(r - 1) to find sum of payments...(so A0 = a, r = (0.1299/12 + 19/20) and n = n.)