Explanation:
To calculate how much money should be deposited today in an account that earns 3% compounded semiannually to accumulate $10,000 in three years, you can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future amount ($10,000 in this case)
P = the principal amount (the amount you want to find)
r = annual interest rate (3% or 0.03)
n = number of times interest is compounded per year (2 for semiannual)
t = the number of years (3 years)
You want to solve for P:
$10,000 = P(1 + 0.03/2)^(2 * 3)
$10,000 = P(1 + 0.015)^(6)
Now, simplify the equation:
$10,000 = P(1.015)^6
$10,000 = P(1.0931640625)
Now, divide both sides by 1.0931640625 to solve for P:
P = $10,000 / 1.0931640625
P ≈ $9,150.05
So, you should deposit approximately $9,150.05 today in the account to accumulate to $10,000 in three years when earning a 3% interest rate compounded semiannually.