Question

You deposit $3000 in an account that earns 4% annual interest compounded continuously. Find the balance of your account after 7 years. Your friend deposits $2800 in an account that earns 5.2% annual interest compounded quarterly. Which account has a greater balance after 7 years?

Answer 1

`~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$3000\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ t=years\dotfill &7 \end{cases} \\\\\\ A = 3000e^(0.04\cdot 7) \implies \boxed{A \approx 3969.39} \\\\[-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 &\$2800\\ r=rate\to 5.2\%\to (5.2)/(100)\dotfill &0.052\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &7 \end{cases}`

`A = 2800\left(1+(0.052)/(4)\right)^(4\cdot 7)\implies A=2800(1.013)^(28) \implies \boxed{A \approx 4019.97} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} 4019.97 ~~ > ~~ 3969.39 \end{array}}~\hfill`