Compare the investment below to an investment of the same principal at the same rate compounded annually. principal: \( \$ 2,000 \), annual interest: \( 9 \% \), interest periods: 12 , number of years: 10 After 10 years, the investment compounded periodically will be worth \( \$ \square \) more than the investment compounded annually. (Round to two decimal places as needed.)

To compare the two investments—one compounded annually and the other compounded monthly—we'll calculate the future value of each and determine the difference after 10 years. ### Given: - **Principal (P):** \$2,000 - **Annual Interest Rate (r):** 9% or 0.09 - **Number of Years (t):** 10 - **Compounding Periods:** - **Annually:** 1 per year - **Monthly:** 12 per year ### Formulas: 1. **Future Value with Annual Compounding:** \[ A_{\text{annual}} = P \times \left(1 + r\right)^t \] 2. **Future Value with Monthly Compounding:** \[ A_{\text{monthly}} = P \times \left(1 + \frac{r}{12}\right)^{12t} \] 3. **Difference:** \[ \text{Difference} = A_{\text{monthly}} - A_{\text{annual}} \] ### Calculations: 1. **Annual Compounding:** \[ A_{\text{annual}} = 2000 \times (1 + 0.09)^{10} = 2000 \times 2.367364 \approx \$4,734.72 \] 2. **Monthly Compounding:** \[ A_{\text{monthly}} = 2000 \times \left(1 + \frac{0.09}{12}\right)^{120} = 2000 \times (1.0075)^{120} \approx 2000 \times 2.45131 \approx \$4,902.62 \] 3. **Difference:** \[ \text{Difference} = 4902.62 - 4734.72 \approx \$167.90 \] ### Conclusion: After 10 years, the investment compounded monthly will be worth **\$167.90** more than the investment compounded annually. **Answer:** After 10 years, the investment compounded periodically is worth **\$167.90** more.