# Math Help - Compounded Quarterly

1. ## 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

2. 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?
. . . $\begin{array}{cc}\text{principal} & P \,=\,1000 \\
\text{amount}& A \,=\,2000 \\
\text{quarterly rate}& \frac{r}{4}\text{ percent} \\
\text{no. of quarters} & n \:=\:32\end{array}$

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

3. ## ok....

Originally Posted by Soroban
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.