Question

is anyone awake to help me?

Answer 1

Question?

Answer 2

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?

Answer 3

Compounding interest?

Answer 4

yes i think so

Answer 5

Is the first payment made at the beginning of the month then, or at the end of it?

Answer 6

idk there nothing more to the question

Answer 7

Let's have \(m\) be the monthly payment.
We'll have \(P_t\) the principle after \(t\) months.\[
P_0 = $9,889
\]\[
P_{t+1} = P_t\times(1+0.235)-m
\]

Answer 8

\[\begin{split}
P_{t}
&= 1.235P_{t-1}-m \\
&= 1.235(1.235P_{t-2}-m)-m\\
&=(1.235)^2P_{t-2}-1.235m-m\\
\end{split}\]

Answer 9

There is a pattern we can find here.

Answer 10

\[\begin{split}
P_{t} &= (1.235)^tP_{t-t}-m\sum_{k=1}^{t}(1.245)^{k-1}\\
&= (1.235)^tP_{0}-m\sum_{k=1}^{t}(1.245)^{k-1}\\
&= (1.235)^t($9,889)-m\sum_{k=1}^{t}(1.245)^{k-1}\\
\end{split}
\]

Answer 11

There is a geometric series here.

Does all of this math looks confusing to you?

Answer 12

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.

Answer 13

\[
m\sum_{k=1}^t(1.245)^k=m\frac{1-(1.245)^t}{1-1.245}
\]

Answer 14

\[
P_t= (1.235)^t($9,889)-m\frac{1-(1.245)^t}{1-1.245}
\]

Answer 15

Now we need to find \(m\) for \(P_{12}\).

Answer 16

Where \(P_{12}=0\) since it needs to be payed off by then.

Answer 17

\[
0=(1.245)^{12}(9889)-m\frac{1-(1.245)^{12}}{1-1.245}
\]Solve for \(m\) here.

Answer 18

interest rate must be divided by 12, other than that it looks good

Answer 19

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

Answer 20

@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

Answer 21

\[
\huge \color{red}{The\ Power\ Of\ Math}
\]

Answer 22

@rsadhvika Thanks for the correction, I often forget that interest rates are annual.

Answer 23

:) it was an amazing derivation u did above !