1. **Problem statement:** We want to find the present deposit amount $P$ that will grow to $20,000$ in $9$ years with an annual interest rate of $5\%$ compounded monthly. 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 compounding periods per year - $t$ is the time in years 3. **Given values:** - $A = 20000$ - $r = 0.05$ - $n = 12$ (monthly compounding) - $t = 9$ 4. **Rearranging the formula to solve for $P$:** $$ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} $$ 5. **Calculate the denominator:** $$ 1 + \frac{0.05}{12} = 1 + 0.0041667 = 1.0041667 $$ $$ nt = 12 \times 9 = 108 $$ $$ \left(1.0041667\right)^{108} \approx 1.551328 $$ 6. **Calculate $P$:** $$ P = \frac{20000}{1.551328} \approx 12889.46 $$ 7. **Answer:** You need to deposit approximately **12889.46** today to have 20000 after 9 years at 5% annual interest compounded monthly.