Question

How much must be deposited today into the following account in order to have $ 40,000 in 6 years for a down payment on a​ house? Assume no additional deposits are made.

An account with annual compounding and an APR of 8​%
​$
nothing should be deposited today.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

Answer 1

`\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill&40000\\ P=\textit{original amount deposited}\dotfill\\ r=rate\to 8\%\to (8)/(100)\dotfill &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annual compounding, thus once} \end{array}\dotfill &1\\ t=years\dotfill &6 \end{cases}`

`\bf 40000=P\left(1+(0.08)/(1)\right)^(1\cdot 6)\implies 40000=P(1.08)^6\implies \cfrac{40000}{1.08^6}=P \\\\\\ 25206.785075\approx P\implies \stackrel{\textit{rounded up}}{25206.79}=P`