1. **State the problem:** We need to find the principal amount $P$ that will grow to $10,000$ in 5 years with an interest rate of 4% 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, - $r$ is the annual interest rate (decimal), - $n$ is the number of times interest is compounded per year, - $t$ is the time in years. 3. **Given values:** - $A = 10000$ - $r = 0.04$ - $n = 4$ (quarterly compounding) - $t = 5$ 4. **Substitute values into the formula:** $$10000 = P \left(1 + \frac{0.04}{4}\right)^{4 \times 5}$$ 5. **Simplify inside the parentheses:** $$10000 = P \left(1 + 0.01\right)^{20}$$ $$10000 = P \left(1.01\right)^{20}$$ 6. **Calculate $1.01^{20}$:** $$1.01^{20} \approx 1.22019$$ 7. **Rewrite the equation:** $$10000 = P \times 1.22019$$ 8. **Solve for $P$ by dividing both sides:** $$P = \frac{10000}{1.22019}$$ 9. **Show cancellation step:** $$P = \frac{10000}{\cancel{1.22019}} \cancel{1.22019}$$ 10. **Calculate $P$:** $$P \approx 8193.80$$ **Final answer:** The principal needed is approximately $8193.80$.