Question

Compare the investment below to an investment of the same principal at the same rate compounded annually (look at picture below for details)

Answer 1

so we have two investments, one compounding annually and another compounding semi-annually, let's check both

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$5000\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus twice} \end{array}\dotfill &2\\ t=years\dotfill &11 \end{cases} \\\\\\ A = 5000\left(1+(0.05)/(2)\right)^(2\cdot 11) \implies \boxed{A \approx 8607.86} \\\\[-0.35em] ~\dotfill`

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$5000\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &11 \end{cases}`
`A = 5000\left(1+(0.05)/(1)\right)^(1\cdot 11) \implies \boxed{A \approx 8551.70} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ semi-annually }{8607.86}~~ - ~~\stackrel{ annually }{8551.70} ~~ \approx ~~ \text{\LARGE 56.16}`