Question

Compare the investment below to an investment of the same principal at the same rate compounded annually.

principal: $8,000, annual interest: 7%, interest periods: 4, number of years: 16
***
After 16 years, the investment compounded periodically will be worth $ more than the investment compounded annually,
(Round to two decimal places as needed.)

Answer 1

`~~~~~~ \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 &\$8000\\ r=rate\to 7\%\to (7)/(100)\dotfill &0.07\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &16 \end{cases} \\\\\\ A = 8000\left(1+(0.07)/(1)\right)^(1\cdot 16) \implies A \approx \boxed{23617.31} \\\\[-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 &\$8000\\ r=rate\to 7\%\to (7)/(100)\dotfill &0.07\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &16 \end{cases}`

`A = 8000\left(1+(0.07)/(4)\right)^(4\cdot 16) \implies \boxed{A \approx 24282.46} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ quarterly }{24282.46}~~ - ~~\stackrel{ annually }{23617.31} ~~ \approx ~~ \text{\LARGE 665.15}`