Question

Suppose that you have $6000 to invest. Which investment yields the greater return over four years: 8.25% compounded quarterly or 8.3% compounded semiannually?

Answer 1

$\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to &\ 6000\\ r=rate\to 8.25\%\to \frac{8.25}{100}\to &0.0825\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\to &4\\ t=years\to &4 \end{cases} \\\\\\ A=6000\left(1+\frac{0.0825}{4}\right)^{4\cdot 4}\implies A=6000(1.020625)^{16}\\\\ -------------------------------$



$\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to &\ 6000\\ r=rate\to 8.3\%\to \frac{8.3}{100}\to &0.083\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semiannually, thus two} \end{array}\to &2\\ t=years\to &4 \end{cases} \\\\\\ A=6000\left(1+\frac{0.083}{2}\right)^{2\cdot 4}\implies A=6000(1.0415)^8$

compare them away.