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.