Question

OWhat interest rate compounded monthly is needed for $500 to triple in 8 years? Show work.

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \stackrel{ tripled }{\$ 1500}\\ P=\textit{original amount deposited}\dotfill &\$500\\ r=rate\to r\%\to (r)/(100)\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &8 \end{cases}`

`1500 = 500\left(1+((r)/(100))/(12)\right)^(12\cdot 8)\implies \cfrac{1500}{500}= \left( 1+\cfrac{r}{1200} \right)^(96) \\\\\\ 3=\left( \cfrac{1200+r}{1200} \right)^(96)\implies \sqrt[96]{3}=\cfrac{1200+r}{1200}\implies 1200\sqrt[96]{3}=1200+r \\\\\\ 1200\sqrt[96]{3}-1200=r\implies \stackrel{ \% }{13.81}\approx r`