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Find how much money needs to be deposited now into an account to obtain $5,200 (Future Value) in 14 years if the interest rate is 4.5% per year compounded annually.

The necessary present value (principal deposit) necessary is $

Round your answer to 2 decimal places

User Hdost
by
8.5k points

2 Answers

3 votes

Answer:

$2,807.86

Explanation:

To determine how much money needs to be deposited in an account to obtain a future value of $5,200 in 14 years given the interest rate is 4.5% per year compounding annually, we can use the future value formula:


\boxed{\begin{array}{l}\underline{\textsf{Future Value Formula}}\\\\FV=P\left(1+(r)/(n)\right)^(nt)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$FV$ is the future value.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount invested.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}

In this case:

  • FV = $5,200
  • r = 4.5% = 0.045
  • n = 1 (compounded annnually)
  • t = 14 years

Substitute the values into the formula:


5200=P\left(1+(0.045)/(1)\right)^(14 \cdot 1)

Solve for P:


5200=P\left(1+0.045\right)^(14)


5200=P\left(1.045\right)^(14)


P=(5200)/((1.045)^(14))


P=(5200)/(1.8519449216...)


P=2807.8588835...


P=2807.86

Therefore, the amount of money that needs to be deposited now is $2,807.86, rounded to two decimal places.

User SpruceMoose
by
7.0k points
4 votes

Answer:

$2807.86

Explanation:

In order to find the present value we can use future value formula:

The future value (FV) formula for compound interest is given by:


\sf FV = PV * (1 + r)^t

Where:

-
\sf FV is the future value (in this case, $5,200),

-
\sf PV is the present value or principal deposit (what we want to find),

-
\sf r is the annual interest rate (4.5% or 0.045 in decimal form), and

-
\sf t is the number of years (14).

We want to rearrange this formula to solve for the present value (
\sf PV):


\sf PV = (FV)/((1 + r)^t)

Now, plug in the values:


\sf PV = (5200)/((1 + 0.045)^(14))

Calculate this expression to find the necessary present value:


\sf PV = (5200)/((1.045)^(14))


\sf PV \approx (5200)/(1.851944922)


\sf PV \approx 2807.858884


\sf PV \approx 2807.86 \textsf{ (in 2 d.p.)}

Therefore, the necessary present value (principal deposit) to obtain $5,200 in 14 years with an annual interest rate of 4.5% compounded annually is approximately $2807.86 (rounded to 2 decimal places).

User Nablex
by
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