To solve this problem, we use the future value formula for compound interest, which is:

$FV = PV \times (1 + r/n)^{n \cdot t}$

Where:

• $FV$ is the future value (here, $15,000),
• $PV$ is the present value (the amount we want to find),
• $r$ is the annual interest rate (9% = 0.09),
• $n$ is the number of compounding periods per year (monthly compounding means $n = 12$),
• $t$ is the time in years (3 years).

Rearranging the formula to solve for $PV$:

$PV = \frac{FV}{(1 + r/n)^{n \cdot t}}$

Substituting the given values:

• $FV = 15,000$,
• $r = 0.09$,
• $n = 12$,
• $t = 3$.

$PV = \frac{15,000}{(1 + 0.09/12)^{12 \cdot 3}}$

Now, calculate step by step:

1. $r/n = 0.09 / 12 = 0.0075$,
2. $1 + r/n = 1 + 0.0075 = 1.0075$,
3. $n \cdot t = 12 \cdot 3 = 36$,
4. $(1 + r/n)^{n \cdot t} = 1.0075^{36}$.

Let’s compute these values:# Given values FV = 15000 # Future Value r = 0.09 # Annual interest rate n = 12 # Compounding periods per year (monthly) t = 3 # Time in years

Calculate (1 + r/n)^(n*t)

compound_factor = (1 + r / n) ** (n * t)

Calculate the present value

PV = FV / compound_factor PV11462.23440923070111462.234409230701successThe amount of money that should be deposited today is approximately $11,462.23.

This ensures the investment grows to $15,000 in three years with 9% annual interest compounded monthly.

Would you like further clarification or have any questions?

Here are 5 related questions:

1. What happens to the amount required if the interest rate increases to 10%?
2. How would daily compounding affect the present value?
3. What is the total interest earned after 3 years?
4. How much should be deposited if the goal is $20,000 instead of $15,000?
5. How does compounding frequency impact the present value calculation?

Tip: Always ensure to check whether interest is compounded annually, monthly, or daily when solving financial problems, as it significantly impacts the result.