Question

You want to retire with $400 000 in the bank and you are able to earn 6% compounded quarterly for the next 25 years. How much money do you have to invest today in order to achieve your goal? a $90 251.78 b $90 251.77 c $90 001.78 d $90 201.77 e $90 001.77

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 400000\\ P=\textit{original amount deposited}\\ r=rate\to 6\%\to (6)/(100)\dotfill &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &25 \end{cases}`

`400000 = P\left(1+(0.06)/(4)\right)^(4\cdot 25) \implies 400000=P(1.015)^(100) \\\\\\ \cfrac{400000}{(1.015)^(100)}=P\implies 90251.78\approx P`

Answer 2

P(1 + .06/4)¹⁰⁰ = $400,000

P = $90,251.78

The correct answer is A.