Question

If you open a bank account with 23,000 and its annual interest is compounded quarterly. What would the interest have to be for the amount to grow to $50,000 in 7 years?

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 50000\\ P=\textit{original amount deposited}\dotfill &\$23000\\ r=rate\to r\%\to (r)/(100)\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &7 \end{cases}`

`50000 = 23000\left(1+( ~~ (r)/(100) ~~ )/(4)\right)^(4\cdot 7) \implies \cfrac{50000}{23000}=\left( 1+\cfrac{r}{400} \right)^(28) \\\\\\ \cfrac{50}{23}=\left( \cfrac{400+r}{400} \right)^(28)\implies \sqrt[28]{\cfrac{50}{23}}=\cfrac{400+r}{400} \\\\\\ 400\sqrt[28]{\cfrac{50}{23}}=400+r\implies 400\sqrt[28]{\cfrac{50}{23}}-400=r\implies \stackrel{ \% }{11.25}\approx r`