You invest \( \$ 800 \) at an annual rate of \( 8 \% \) for 12 years. How much more interest would you earn with compound interest than with simple interest for (a) the whole 12 years; and (b) in just year 7 by itself?

To find out how much more interest you would earn with compound interest compared to simple interest, let’s calculate both types of interest. 1. **Simple Interest**: The formula for simple interest is \( I = P \cdot r \cdot t \), where \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years. \[ I = 800 \cdot 0.08 \cdot 12 = 800 \cdot 0.96 = 768 \] So, the total simple interest earned over 12 years is \( \$ 768 \). 2. **Compound Interest**: The formula for compound interest is \( A = P(1 + r)^t \), where \( A \) is the amount of money accumulated after n years, including interest. \[ A = 800(1 + 0.08)^{12} \approx 800(2.51817) \approx 2014.54 \] The total compound interest earned is \( A - P = 2014.54 - 800 \approx 1214.54 \). Now, let's calculate the difference in interest earned: \[ \text{Difference} = 1214.54 - 768 = 446.54 \] So, you would earn **about \( \$ 446.54 \)** more with compound interest than with simple interest over the entire 12 years. **(b)** To find the interest earned in just year 7 with compound interest, we can calculate the total amount at the end of year 6, then find the amount at the end of year 7, and subtract the two: \[ A_{6} = 800(1 + 0.08)^{6} \approx 800(1.58687) \approx 1269.50 \] \[ A_{7} = 800(1 + 0.08)^{7} \approx 800(1.71382) \approx 1371.06 \] The interest earned during year 7 is: \[ I_{7} = A_{7} - A_{6} = 1371.06 - 1269.50 \approx 101.56 \] For simple interest during year 7, it is the same as each year: \[ I_{7, \text{simple}} = 800 \cdot 0.08 = 64 \] The difference in interest for year 7: \[ \text{Difference}_{7} = 101.56 - 64 \approx 37.56 \] Thus, you would earn **about \( \$ 37.56 \)** more with compound interest than with simple interest in just year 7.