Question

At what interest rate(to the nearest hundredth of a percent) compounded annually will money in savings double in five years

Answer 1

if the amount of money is say $1, for that to double it'll become $2, so let's use those values.

`\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill&\$2\\ P=\textit{original amount deposited}\dotfill &\$1\\ r=rate\to r\%\to (r)/(100)\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &5 \end{cases}`

`\bf 2=1\left(1+(r)/(1)\right)^(1\cdot 5)\implies 2=(1+r)^5\implies \sqrt[5]{2}=1+r \\\\\\ \sqrt[5]{2}-1=r\implies 0.148698\approx r\implies \stackrel{\textit{converting to \%}}{0.148698\cdot 100\approx r\%}\implies 14.8698\approx r \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{rounded up}}{14.87=r}~\hfill`