Question

Tia is investing $2500 that she would like to grow to $6000 in 10 years. At what annual interest rate, compounded quarterly, must Tia invest her money? Round your answer to two decimal places.

Answer 1

$\bf \qquad \textit{Compound Interest Earned Amount}\\ A=P\left(1+\frac{r}{n}\right)^{nt} \\ \quad \\ \begin{cases} A=\textit{current amount}\to &6,000\\ P=\textit{original amount deposited}\to &\ 2,500\\ r=rate\to r\%\to \frac{r}{100}\to &0.0r\\ n=\textit{times it compounds per year, quarterly}\to &4\\ t=years\to &10 \end{cases} \\ \quad \\ meaning \\ \quad \\ 6,000=2,500\left(1+\frac{r}{4}\right)^{4\cdot 10}$

solve for "r"

notice, the "r" value, goes in the equation as a decimal, so 5% is really 5/100 or 0.05 25% is 0.25 and so on

so... the "r" that you'll get, will be a decimal value, multiply that by 100, and you'll get the percentage notation of it

$\bf 6,000=2,500\left(1+\frac{r}{4}\right)^{4\cdot 10}\implies\cfrac{6000}{2500}=\left(1+\frac{r}{4}\right)^{40} \\ \quad \\ \cfrac{12}{5}=\left(1+\frac{r}{4}\right)^{40}\impliedby \textit{now, taking root }\sqrt[40]{\qquad } \\ \quad \\ \sqrt[40]{\cfrac{12}{5}}=\sqrt[40]{\left(1+\frac{r}{4}\right)^{40}}\implies \sqrt[40]{\cfrac{12}{5}}=1+\cfrac{r}{4} \\ \quad \\ \sqrt[40]{\cfrac{12}{5}}-1=\cfrac{r}{4}\implies 4\left( \sqrt[40]{\cfrac{12}{5}}-1 \right)=r \\ \quad$
$\bf 0.0885119586095070592=r\implies 0.0885\cdot 100\implies 8.85\%=r$