How much money should be deposited today in an account that earns 6.5% compounded monthly so that it will accumlate to $8,000.00 in three years?

monthly rate = .065/12 = .00541666..

so
x(1.00541666..)^36 = 8000
x = 8000/1.005416666^36 = 6586.14

To determine the amount of money that should be deposited today, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the principal amount (the amount to be deposited today)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years

In this case, we have:
A = $8,000.00 (the desired accumulated amount)
r = 0.065 (6.5% expressed as a decimal)
n = 12 (since it is compounded monthly)
t = 3 (three years)

Let's plug these values into the formula and solve for P:

$8,000.00 = P(1 + 0.065/12)^(12*3)

Now let's simplify and solve the equation step by step:

$8,000.00 = P(1 + 0.00541667)^(36)
$8,000.00 = P(1.00541667)^(36)

To find P, divide both sides of the equation by (1.00541667)^(36):

P = $8,000.00 / (1.00541667)^(36)

Using a calculator or spreadsheet, evaluate (1.00541667)^(36), which equals approximately 1.231112511.

P ≈ $8,000.00 / 1.231112511

P ≈ $6,500.07

Therefore, approximately $6,500.07 should be deposited today in the account to accumulate to $8,000.00 in three years at an interest rate of 6.5% compounded monthly.