anonymous
  • anonymous
is anyone awake to help me?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Question?
anonymous
  • anonymous
Johann owes $9,889 on a credit card with a 23.5% interest rate. What is the monthly payment he should make to pay off this debt in 12 months, assuming he does not charge any more purchases with the card?
anonymous
  • anonymous
Compounding interest?

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anonymous
  • anonymous
yes i think so
anonymous
  • anonymous
Is the first payment made at the beginning of the month then, or at the end of it?
anonymous
  • anonymous
idk there nothing more to the question
anonymous
  • anonymous
Let's have \(m\) be the monthly payment. We'll have \(P_t\) the principle after \(t\) months.\[ P_0 = $9,889 \]\[ P_{t+1} = P_t\times(1+0.235)-m \]
anonymous
  • anonymous
\[\begin{split} P_{t} &= 1.235P_{t-1}-m \\ &= 1.235(1.235P_{t-2}-m)-m\\ &=(1.235)^2P_{t-2}-1.235m-m\\ \end{split}\]
anonymous
  • anonymous
There is a pattern we can find here.
anonymous
  • anonymous
\[\begin{split} P_{t} &= (1.235)^tP_{t-t}-m\sum_{k=1}^{t}(1.245)^{k-1}\\ &= (1.235)^tP_{0}-m\sum_{k=1}^{t}(1.245)^{k-1}\\ &= (1.235)^t($9,889)-m\sum_{k=1}^{t}(1.245)^{k-1}\\ \end{split} \]
anonymous
  • anonymous
There is a geometric series here. Does all of this math looks confusing to you?
anonymous
  • anonymous
If so, then there is a set formula you should have learned that you should be using. In a way I'm deriving that formula.
anonymous
  • anonymous
\[ m\sum_{k=1}^t(1.245)^k=m\frac{1-(1.245)^t}{1-1.245} \]
anonymous
  • anonymous
\[ P_t= (1.235)^t($9,889)-m\frac{1-(1.245)^t}{1-1.245} \]
anonymous
  • anonymous
Now we need to find \(m\) for \(P_{12}\).
anonymous
  • anonymous
Where \(P_{12}=0\) since it needs to be payed off by then.
anonymous
  • anonymous
\[ 0=(1.245)^{12}(9889)-m\frac{1-(1.245)^{12}}{1-1.245} \]Solve for \(m\) here.
rsadhvika
  • rsadhvika
interest rate must be divided by 12, other than that it looks good
Directrix
  • Directrix
I cannot vouch for this online payment calculator but you might want to take a look. http://www.bankrate.com/calculators/credit-cards/credit-card-payoff-calculator.aspx
rsadhvika
  • rsadhvika
@Directrix thats correct actually, wolfram gives the same :- http://www.wolframalpha.com/input/?i=9889%281%2B.235%2F12%29%5E12+%3D+x%28%281-%281%2B.235%2F12%29%5E12%29%2F%281-%281%2B.235%2F12%29%29%29
anonymous
  • anonymous
\[ \huge \color{red}{The\ Power\ Of\ Math} \]
anonymous
  • anonymous
@rsadhvika Thanks for the correction, I often forget that interest rates are annual.
rsadhvika
  • rsadhvika
:) it was an amazing derivation u did above !

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