Question

You want to set up a retirement account. You are going to put $25,000 into the account. How
long will it take for each account to double in value.
1) An account offering 5.3% interest per year
compounded quarterly.
2) An account offering 7.2% interest per year
compounded monthly.

Answer 1

$~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill &\stackrel{double~25000}{\ 50000~~~~}\\ P=\textit{original amount deposited}\dotfill &\ 25000\\ r=rate\to 5.3\%\to \frac{5.3}{100}\dotfill &0.053\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years \end{cases}$

$50000=25000\left(1+\frac{0.053}{4}\right)^{4\cdot t}\implies \cfrac{50000}{25000}=(1.01325)^{4t}\implies 2=(1.01325)^{4t} \\\\\\ \log(2)=\log[(1.01325)^{4t}]\implies \log(2)=t\log(1.01325^4) \\\\\\ \cfrac{\log(2)}{\log(1.01325^4)}=t\implies 13\approx t \\\\[-0.35em] ~\dotfill$

$~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill &\stackrel{double~25000}{\ 50000~~~~}\\ P=\textit{original amount deposited}\dotfill &\ 25000\\ r=rate\to 7.2\%\to \frac{7.2}{100}\dotfill &0.072\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years \end{cases}$

$50000=25000\left(1+\frac{0.072}{12}\right)^{12\cdot t}\implies \cfrac{50000}{25000}=(1.006)^{12t}\implies 2=(1.006)^{12t} \\\\\\ \log(2)=\log[(1.006)^{12t}]\implies \log(2)=t\log(1.006^{12}) \\\\\\ \cfrac{\log(2)}{\log(1.006^{12})}=t\implies \stackrel{\textit{about 9 years and 252 days }}{9.7\approx t}$