1. **State the problem:** Stephy invested a portion of her prize money at an interest rate of 3.95% compounded semi-annually. After 4 years, the investment grew to 263106.90. We need to find the initial amount she invested. 2. **Identify the formula:** The compound interest formula is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $A$ is the amount after time $t$ - $P$ is the principal (initial investment) - $r$ is the annual interest rate (decimal) - $n$ is the number of compounding periods per year - $t$ is the number of years 3. **Plug in known values:** - $A = 263106.90$ - $r = 0.0395$ - $n = 2$ (semi-annually) - $t = 4$ 4. **Calculate the expression inside the parentheses:** $$1 + \frac{0.0395}{2} = 1 + 0.01975 = 1.01975$$ 5. **Calculate the exponent:** $$nt = 2 \times 4 = 8$$ 6. **Rewrite the formula to solve for $P$:** $$P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} = \frac{263106.90}{1.01975^8}$$ 7. **Calculate $1.01975^8$:** $$1.01975^8 \approx 1.171659$$ 8. **Calculate $P$:** $$P = \frac{263106.90}{1.171659} \approx 224572.15$$ **Final answer:** Stephy initially invested approximately $224572.15.