Manually calculate the compound interest on an investment of \( \$ 7,500 \) at \( 6 \% \) interest, compounded semiannually, for 1 year.

To calculate compound interest, the formula we use is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (initial investment). - \( r \) 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 for in years. In this case: - \( P = 7,500 \) - \( r = 0.06 \) (which is 6% expressed as a decimal) - \( n = 2 \) (since interest is compounded semiannually) - \( t = 1 \) Substituting these values into the formula gives: \[ A = 7500 \left(1 + \frac{0.06}{2}\right)^{2 \times 1} \] Calculating it step-by-step: 1. Calculate \( \frac{0.06}{2} = 0.03 \). 2. Add 1: \( 1 + 0.03 = 1.03 \). 3. Calculate exponent: \( 2 \times 1 = 2 \). 4. Calculate \( (1.03)^2 = 1.0609 \). 5. Finally calculate \( A = 7500 \times 1.0609 \approx 7956.75 \). The total amount after 1 year is approximately \( \$ 7,956.75 \). To find the compound interest earned, subtract the principal: \[ \text{Compound Interest} = A - P = 7956.75 - 7500 = 456.75 \] The compound interest earned is approximately \( \$ 456.75 \).