How much money should be deposited today in an account that earns 7% compounded semiannually so that it will accumulate to $9000 in 3 years?
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, which is $9000
P = the principal amount (the amount of money deposited today)
r = the annual interest rate (7% or 0.07)
n = the number of times the interest is compounded per year (2 times for semiannual compounding)
t = the number of years the money is invested for (3 years)
Plugging in the values we have:
9000 = P(1 + 0.07/2)^(2*3)
9000 = P(1 + 0.035)^6
9000 = P(1.035)^6
9000 = P(1.227615)
P = 9000 / 1.227615
P ≈ $7325.72
Therefore, approximately $7325.72 should be deposited today in the account to accumulate to $9000 in 3 years with an annual interest rate of 7% compounded semiannually.