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 7%how much should be deposited today? $(Do not round until the final answer. Then round to the nearest cent as needed.)

Answer 1

Step-by-step explanation

The formula to calculate the compound interest is:

`\begin{gathered} A=P(1+(r)/(n))^(nt) \\ \text{ Where } \\ A\text{ is the total amount at the end of t years.} \\ P\text{ is the initial amount deposited.} \\ r\text{ is the interest rate.} \\ t\text{ is the time.} \end{gathered}`

From the word problem, we have:

`\begin{gathered} A=40,000 \\ P=? \\ r=7\%=(7)/(100)=0.07 \\ n=1\Rightarrow\text{ It was compounded once in a year.} \\ t=6 \end{gathered}`

Then, we can replace the above values in the compound interest formula and solve for P.

`\begin{gathered} A=P(1+(r)/(n))^(nt) \\ 40,000=P(1+(0.07)/(1))^((1)(6)) \\ 40,000=P(1+0.07)^6 \\ 40,000=P(1.07)^6 \\ \text{ Divide by }1.07^6\text{ from both sides} \\ (40,000)/(1.07^6)=(P(1.07)^6)/(1.07^6) \\ 26,653.69\approx P \\ \text{ The symbol is read 'approximately'.} \end{gathered}`Answer

You should deposit today $26,653.69.