How much money would you have to invest at 8% compounded semiannually so that the total investment has a value of $2,280 after one year?
To determine the amount of money you would have to invest at 8% compounded semiannually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment ($2,280 in this case)
P = the initial principal investment (the amount we want to find)
r = the annual interest rate (8% or 0.08)
n = the number of times interest is compounded per year (semiannually, so 2)
t = the number of years (1 year in this case)
Now, let's plug in the given values into the formula and solve for P:
2,280 = P(1 + 0.08/2)^(2*1)
Simplifying the equation, we have:
2,280 = P(1 + 0.04)^2
Now, we can further simplify:
2,280 = P(1.04)^2
To isolate P, divide both sides of the equation by (1.04)^2:
P = 2,280 / (1.04)^2
Using a calculator, we find:
P ≈ 2,280 / 1.0816
P ≈ $2,109.47
Therefore, you would need to invest approximately $2,109.47 at 8% compounded semiannually to have a total investment value of $2,280 after one year.
P(1 + .08/2)^(2*1) = 2280
P = 2108.00