Question

Compare the investment below to an investment of the same principal at the same rate compounded annually. ​principal: ​$7,000​, annual​ interest: 9​%, interest​ periods:4 ​, number of​ years:12 Question content area bottom Part 1 After 12 ​years, the investment compounded periodically will be worth ​$ enter your response here more than the investment compounded annually.

Answer 1

`~~~~~~ \stackrel{\textit{\LARGE Quarterly}}{\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 &\$7000\\ r=rate\to 9\%\to (9)/(100)\dotfill &0.09\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &12 \end{cases}`

`A = 7000\left(1+(0.09)/(4)\right)^(4\cdot 12) \implies A=7000(1.0225)^(48)\implies \boxed{A \approx 20367.48} \\\\[-0.35em] ~\dotfill`

`~~~~~~ \stackrel{\textit{\LARGE Annually}}{\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 &\$7000\\ r=rate\to 9\%\to (9)/(100)\dotfill &0.09\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &12 \end{cases}`

`A = 7000\left(1+(0.09)/(1)\right)^(1\cdot 12)\implies A=7000(1.09)^(12) \implies \boxed{A \approx 19688.65} \\\\[-0.35em] ~\dotfill\\\\ 20367.48~~ - ~~19688.65 ~~ \approx ~~ \text{\LARGE 678.83}`