Final answer:
To accumulate $17,000 at an 8.5% interest rate compounded semiannually for 10 years, one would need to invest approximately $7,971.70 now. This is calculated using the compound interest formula rearranged to solve for the principal amount.
Step-by-step explanation:
To determine the amount to be invested now to accumulate $17,000 at an 8.5% interest rate compounded semiannually for 10 years, we can utilize the formula for compound interest:
A = P(1 + r/n)nt
Where:
- A is the future value of the investment/loan, including interest.
- P is the principal investment amount (the initial deposit or loan amount).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested or borrowed for, in years.
In this case, we know that A = $17,000, r = 8.5% or 0.085, n = 2 (since the interest is compounded semiannually), and t = 10. We need to solve for P, which is the principal amount to invest.
Rearranging the formula to solve for P gives:
P = A / (1 + r/n)nt
P = $17,000 / (1 + 0.085/2)(2)(10)
Using a calculator, we find:
P = $17,000 / (1 + 0.0425)20
P = $17,000 / (1.0425)20
P = $17,000 / 2.1336 (approximately)
P = $7,971.70 (approximately)
Therefore, approximately $7,971.70 should be invested now to get $17,000 after 10 years, with the given compound interest conditions.