Manually calculate the compound interest on an investment of \( \$ 7,500 \) at \( 6 \% \) interest, compounded semiannually, for 1 year. Select one: a. \( \$ 456.75 \) b. \( \$ 7,956.75 \) o c. \( \$ 7,950.00 \) o d. \( \$ 450.00 \)

To calculate the compound interest, we use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (\$7,500). - \( r \) is the annual interest rate (decimal) (6% = 0.06). - \( n \) is the number of times that interest is compounded per year (2 for semiannual). - \( t \) is the number of years the money is invested (1 year). Plugging in the values: \[ A = 7500 \left(1 + \frac{0.06}{2}\right)^{2 \times 1} \] Calculating: \[ A = 7500 \left(1 + 0.03\right)^{2} = 7500 \left(1.03\right)^{2} \] \[ A = 7500 \times 1.0609 \approx 7956.75 \] Now, the compound interest earned is: \[ A - P = 7956.75 - 7500 = 456.75 \] So, the answer is: **a. \( \$ 456.75 \)** Now, for some added fun facts! Did you know that the concept of compound interest dates back to ancient civilizations? The Babylonians were among the first to use it around 3000 BC, tracking loans and their repayments over time, ensuring that the borrower paid back more than they borrowed! If you want to dig deeper into the fascinating world of finance, try reading "The Richest Man in Babylon" by George S. Clason. It's a classic that offers timeless financial wisdom through entertaining parables set in ancient Babylon—perfect for anyone looking to boost their money management skills!