The principle required for gathering 5000 over 10 years at 10% compounded semiannually can be calculated using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($5000)
P = the principal investment amount (unknown)
r = the annual interest rate (10% or 0.10)
n = the number of times interest is compounded per year (2 for semiannual compounding)
t = the number of years the money is invested (10 years)
Substitute the given values into the formula and solve for P:
5000 = P(1 + 0.10/2)^(2*10)
5000 = P(1 + 0.05)^20
5000 = P(1.05)^20
P = 5000 / (1.05)^20
P ≈ $1605.73
Therefore, the principal required to gather $5000 over 10 years at 10% compounded semiannually is approximately $1605.73.
