Final answer:
To find out how much money must be invested to earn $500 interest over 6 months at a 10% interest rate compounded quarterly, one must use the compound interest formula, account for the number of compounding periods, and solve for the principal amount.
Step-by-step explanation:
To determine how much money needs to be invested to earn $500 in interest over 6 months with a quarterly compounded interest rate of 10%, we use the formula for compound interest, which differs from that used for simple interest. The formula for compound interest is A = P(1 + r/n)(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per unit t, and t is the time the money is invested for, in years.
In this scenario, the end amount A is the principal plus $500 in interest. To find the principal P, we need to solve for P in the formula A = P(1 + r/n)(nt), taking into consideration the given interest rate (10% or 0.1), the time period (0.5 years for 6 months), and that it is compounded quarterly (so n = 4).
After setting up the equation with these values and solving for P, we can determine the initial investment required to achieve the $500 interest. It's important to note that compound interest differs from simple interest, where the calculation is more straightforward as it doesn't consider the compounding effect.