We are dealing with compound interest here, specifically with interest compounded monthly. To find out how much we need to deposit now to have $20,000 in 10 years at 5% interest compounded monthly, we will use the Periodic Compound Interest formula:

$PN = P_0 \left(1 + \frac{f}{k}\right)^{Nk}$

Where:

• $PN$ is the balance after $N$ years (in our case, $20,000),
• $P_0$ is the initial principal or the amount to deposit (what we are trying to find),
• $f$ is the annual interest rate (5% or 0.05),
• $k$ is the number of compounding periods per year (monthly compounding means $k = 12$),
• $N$ is the number of years (10 years).

Let's plug in the values and solve for $P_0$.

1. $PN = 20000$,
2. $f = 0.05$,
3. $k = 12$,
4. $N = 10$.

The formula becomes:

$20000 = P_0 \left(1 + \frac{0.05}{12}\right)^{10 \times 12}$

Now we can solve for $P_0$.The amount you would need to deposit now in order to have $20,000 in 10 years, assuming a 5% interest rate compounded monthly, is $12,143.22 (rounded to two decimal places).

Do you want more details or have any other questions?

Here are 5 related questions to expand on this topic:

1. How would the deposit change if the interest was compounded quarterly instead of monthly?
2. What would happen if the interest rate was 6% instead of 5%?
3. How much total interest would you earn over the 10 years?
4. How does continuous compounding compare to monthly compounding in this scenario?
5. What if you wanted to reach $30,000 instead of $20,000 in 10 years?

Tip: Compounding more frequently generally results in more interest earned over time, as each compounding period builds on the last one.