Question

Find how long it takes a $ 900.00 investment to earn $ 120.00 in interest if it is invested at 6% compounded annually.

Answer 1

so, we know the earned interest is 120, and we know the original deposited amount is 900 bucks, so, the accumulated amount will then be 900 + 120 or 1020 bucks.


`\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\to &\$1020\\ P=\textit{original amount deposited}\to &\$900\\ r=rate\to 6\%\to (6)/(100)\to &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\to &1\\ t=years \end{cases}`


`\bf 1020=900\left(1+(0.06)/(1)\right)^(1\cdot t)\implies \cfrac{1020}{900}=(1.06)^t\implies \cfrac{102}{90}=(1.06)^t \\\\\\ log\left((102)/(90) \right)=log(1.06^t)\implies log\left((102)/(90) \right)=t\cdot log(1.06) \\\\\\ \cfrac{log\left((102)/(90) \right)}{log(1.06)}=t\implies 20.92 \approx t`