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+(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 (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+(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+(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 (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+(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}`