1. **State the problem:** Sarah wants to find the initial investment amount (principal $P$) that will grow to $1030$ in $5$ years with an interest rate of $5.3\%$ compounded quarterly. 2. **Formula used:** 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 time in years 3. **Identify values:** - $A = 1030$ - $r = 0.053$ - $n = 4$ (quarterly compounding) - $t = 5$ 4. **Substitute values into the formula:** $$1030 = P \left(1 + \frac{0.053}{4}\right)^{4 \times 5}$$ 5. **Simplify inside the parentheses:** $$1 + \frac{0.053}{4} = 1 + 0.01325 = 1.01325$$ 6. **Calculate the exponent:** $$4 \times 5 = 20$$ 7. **Rewrite the equation:** $$1030 = P \times (1.01325)^{20}$$ 8. **Calculate $(1.01325)^{20}$:** $$ (1.01325)^{20} \approx 1.29744 $$ 9. **Solve for $P$:** $$P = \frac{1030}{1.29744}$$ 10. **Show cancellation step:** $$P = \frac{1030}{\cancel{1.29744}} \times \frac{\cancel{1}}{1} = \frac{1030}{1.29744}$$ 11. **Calculate $P$:** $$P \approx 793.68$$ 12. **Round to nearest dollar:** $$P \approx 794$$ **Final answer:** Sarah needs to invest approximately **794** dollars to reach 1030 dollars in 5 years with 5.3% interest compounded quarterly.