Final answer:
To calculate the annual interest rate that yields $78 from a $400 investment in half a year, we use the formula I = PRT. By solving for the rate (R), we find that the annual interest rate is 39%.
Step-by-step explanation:
Calculating the interest rate required for a $400 investment to yield $78 in half a year involves understanding the concepts of simple interest and annual percentage rate (APR). In this case, the interest earned is directly proportional to the principal, the interest rate, and the time period. To find the annual interest rate from the interest accrued over a half-year period, we use the simple interest formula I = PRT, where I stands for interest, P stands for principal, R stands for the rate (annual percentage rate), and T stands for time in years.
Using the information given, we substitute $78 for I, $400 for P, and 0.5 for T (as the time period is half a year), to get $78 = $400 × R × 0.5. Solving for R, which represents our annual interest rate, we find that $78 = $200 × R. Therefore, R = $78 / $200, which gives us 0.39 or 39% as the annual interest rate.
Additionally, it's important to note that if interest rates were different or more time periods were involved, the calculations and the resulting annual percentage rate could vary significantly. Interest rates have a profound effect on the value and yield of financial instruments such as bonds and savings accounts. As exemplified in the reference information, when market interest rates increase, the price of pre-existing bonds decreases to yield a competitive rate.