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Math Help - Compounded Quarterly

  1. #1
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    Compounded Quarterly

    Sarah's investment doubled from $1000 to $2000 over 8 years. He knows that the interest was compounded quarterly. What annual rate did he get on his investment?

    Is the formula Quarterly = p(1 + r/4)^4
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  2. #2
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    Hello, magentarita!

    You were quite close . . .


    Sarah's investment doubled from $1000 to $2000 over 8 years.
    He knows that the interest was compounded quarterly.
    What annual rate did he get on his investment?
    . . . \begin{array}{cc}\text{principal} & P \,=\,1000 \\<br />
\text{amount}& A \,=\,2000 \\<br />
\text{quarterly rate}& \frac{r}{4}\text{ percent} \\<br />
\text{no. of quarters} & n \:=\:32\end{array}

    We have: . 1000\left(1 + \frac{r}{4}\right)^{32} \:=\:2000 \quad\Rightarrow\quad \left(1 + \frac{r}{4}\right)^{32} \:=\:2

    Take the 32^{nd}\text{ root: }\;1 + \frac{r}{4} \:=\:2^{\frac{1}{32}} \quad\Rightarrow\quad \frac{r}{4} \:=\:2^{\frac{1}{32}} - 1


    Therefore: . r \;=\;4\left[2^{\frac{1}{32}}-1\right] \;=\;0.087588595... \;\approx\;8.76\%

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  3. #3
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    ok....

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    You were quite close . . .

    . . . \begin{array}{cc}\text{principal} & P \,=\,1000 \\ \text{amount}& A \,=\,2000 \\
    \text{quarterly rate}& \frac{r}{4}\text{ percent} \\
    \text{no. of quarters} & n \:=\:32\end{array}" alt="
    \text{amount}& A \,=\,2000 \\
    \text{quarterly rate}& \frac{r}{4}\text{ percent} \\
    \text{no. of quarters} & n \:=\:32\end{array}" />

    We have: . 1000\left(1 + \frac{r}{4}\right)^{32} \:=\:2000 \quad\Rightarrow\quad \left(1 + \frac{r}{4}\right)^{32} \:=\:2

    Take the 32^{nd}\text{ root: }\;1 + \frac{r}{4} \:=\:2^{\frac{1}{32}} \quad\Rightarrow\quad \frac{r}{4} \:=\:2^{\frac{1}{32}} - 1


    Therefore: . r \;=\;4\left[2^{\frac{1}{32}}-1\right] \;=\;0.087588595... \;\approx\;8.76\%
    Great stuff from you as always.
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