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So here is the problem:Luke is borrowing $\$10{,}000$ from the bank. The bank offers him a choice between two $10$-year payment plans:

Plan 1 Luke's debt accumulates $10\%$ annual interest which compounds quarterly. Luke pays off half his balance after $5$ years, and the rest at the end of the $10$ years.

Plan 2 Luke's debt accumulates $10\%$ annual interest which compounds annually. Luke pays off his full balance at the end of the $10$ years.

What is the (positive) difference between Luke's total payments under Plan 1 and his total payments under Plan 2? Round to the nearest dollar.

Now, the second plan's payment amount is $10,000 * (1 +\frac{1}{10})^{10}$ which is around $25937$.

In the first plan, he will pay off $\frac{1}{2} *10000 (1+\frac{1}{10})^{20}$ after five years. The remaining amount will continue to be compounded, which he pays off after another five years, which is $\frac{1}{2} * 10000(1+\frac{1}{10})^{40}$ which is around 259933. The difference between the plans is a lot. Where did I go wrong?

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  • $\begingroup$ Please read the tag description before choosing the tag. $\endgroup$ – N. F. Taussig Apr 27 '18 at 14:54
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    $\begingroup$ If the annual rate for plan 1 is 10%, then the quarterly factor should be $(1 + \frac{0.10}{4})$, shouldn't it? And the number of quarters in $5$ years is $20$. $\endgroup$ – MPW Apr 27 '18 at 14:54
  • $\begingroup$ Please don't use MathJax for normal text. MathJax is for objects in mathematical discourse. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Apr 27 '18 at 14:56
  • $\begingroup$ Oh.... I see what you mean. So my mistake was thinking the annual rate is the same as the quarterly rate/ $\endgroup$ – TheLeogend Apr 27 '18 at 14:56
  • $\begingroup$ Is MathJax like Latex? Where you put dollar signs around the numbers? $\endgroup$ – TheLeogend Apr 27 '18 at 14:59
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On calculating the first plan, $\frac 12*10000(1+\frac{1}{10})^{20}$ should be replaced by $10000(1+\frac{2.5}{100})^{20}$ because in the FIRST FIVE years the interest is being compounded quarterly on the complete amount. The interest rate is 10% per year and not 10% per quarter.

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