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I am having difficulty with the following question.

Leona bought a 1-year 10,000 certificate of deposit that paid an interest at an annual rate of 8 percent compounded semi-annually . What was the total amount of interest paid on this certificate at maturity

Any suggestions on how I could solve this problem ?

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up vote 2 down vote accepted

If the nominal yearly rate is $8\%$, and we are compounding every half year, then the half-yearly rate is $\frac{8}{2}\%$, that is, $0.04$. So if we have $A$ in the bank, ater a half-year we have $A+A(0.04)$, that is, $A(1.04)$. So our investment gets multiplied by $1.04$ every half-year.

In the first half-year, it grows to $10000(1.04)$. In the next half-year, that gets multiplied by $1.04$, so at the end of the year we have $10000(1.04)(1.04)$, or more simply $10000(1.04)^2$.

But $10000$ of this is principal which is being repaid. The interest paid on the certificate at the end of the year is therefore $10000(1.04)^2-10000$. The result should be $816$.

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What does compounding mean here ? How did you get 1.04 ? – MistyD Aug 29 '12 at 13:00
When the nominal rate per year is $r$, compounded $k$ times a year (popular: $k=12$), then the interest rate per compounding period is $\frac{r}{k}$. Here $k=2$ and $r=8\%=0.08$. So interest rate per half-year is $0.04$. In a half year, $A$ dollars grow to $A(1.04)$ dollars. In the next half year, you get interest not only on your principla, but on the interest earned in the first half year. In your problem, after a half-year you have $10400$. In the second half of the year, you get interest not only on the $10000$, but also on the $400$. Interest on interest, that's compounding. – André Nicolas Aug 29 '12 at 14:11

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