1. **State the problem:** Sophia wants to find the initial investment amount (principal $P$) needed to reach a future value $A = 1450$ in 14 years with an interest rate of 3.4% compounded monthly. 2. **Formula used:** The compound interest formula is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $A$ is the amount of money accumulated after $t$ years, including interest. - $P$ is the principal (initial investment). - $r$ is the annual interest rate (decimal). - $n$ is the number of times interest is compounded per year. - $t$ is the time the money is invested for in years. 3. **Given values:** - $A = 1450$ - $r = 0.034$ - $n = 12$ (monthly compounding) - $t = 14$ 4. **Rearrange the formula to solve for $P$:** $$P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$$ 5. **Calculate the denominator:** $$1 + \frac{r}{n} = 1 + \frac{0.034}{12} = 1 + 0.0028333 = 1.0028333$$ $$nt = 12 \times 14 = 168$$ 6. **Calculate the compound factor:** $$\left(1.0028333\right)^{168} \approx 1.5683$$ 7. **Calculate $P$:** $$P = \frac{1450}{1.5683}$$ 8. **Simplify with cancellation:** $$P = \frac{1450}{\cancel{1.5683}}$$ 9. **Final calculation:** $$P \approx 924.88$$ 10. **Answer:** Sophia needs to invest approximately **925** dollars to reach 1450 dollars in 14 years with 3.4% interest compounded monthly.