Question: Suppose $15,000$ is deposited into an account paying $6.5\%$ interest, which is compounded annually. How much money is in the account after five years, if no withdrawals or additional deposits are made?

1. **State the problem:** We want to find the amount of money in an account after $5$ years with an initial deposit of $15,000$ and an annual interest rate of $6.5\%$ compounded annually. 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 deposit), - $r$ is the annual interest rate (decimal), - $n$ is the number of times interest is compounded per year, - $t$ is the number of years. 3. **Identify values:** - $P = 15000$ - $r = 0.065$ (since $6.5\% = 0.065$) - $n = 1$ (compounded annually) - $t = 5$ 4. **Substitute values into formula:** $$A = 15000 \left(1 + \frac{0.065}{1}\right)^{1 \times 5} = 15000 \left(1 + 0.065\right)^5 = 15000 \times 1.065^5$$ 5. **Calculate $1.065^5$:** $$1.065^5 = 1.065 \times 1.065 \times 1.065 \times 1.065 \times 1.065 = 1.37009$$ (rounded to 5 decimal places) 6. **Calculate final amount:** $$A = 15000 \times 1.37009 = 20551.35$$ 7. **Interpretation:** The amount in the account after $5$ years is approximately $20,551.35$. 8. **Compare with options:** The closest option is $20,551.30$. **Final answer:** $20,551.30$