Final answer:
To find the present value of a $1000 investment at an 8% annual interest rate compounded semi-annually over 5 years, we use the compound interest formula rearranged to solve for the principal.
Step-by-step explanation:
To calculate the present value of the $1000 investment, we will rearrange the compound interest formula to solve for the principal (P). According to the given formula A = P(1 + r/n)^(nt), rearranging for P gives us P = A / [(1 + r/n)^(nt)]. In this scenario, A is the future value, which we know to be $1000, r is the annual interest rate of 0.08 (after converting 8% into a decimal), n is the number of times interest is compounded per year, which is 2 (since it's compounded every 6 months), and t is the time in years that the money has been invested, which is 5 years.
Now, plug in the values: P = 1000 / [(1 + 0.08/2)^(2*5)]. Breaking down the calculation: (1 + 0.08/2) = 1.04, (1.04)^(2*5) or (1.04)^10 ≈ 1.48024. Therefore, P ≈ 1000 / 1.48024 ≈ $675.56 when rounded to the nearest cent.
The initial principal amount invested was approximately $675.56. To confirm the future value, if we took $675.56 and applied the compound interest formula, it should grow to $1000 over the 5 years at an 8% interest rate compounded semi-annually. This serves as a check of our work.