To solve this problem, we need to determine the monthly deposit Emilia and Liam should make in order to accumulate $200,000 over 12 years, with an interest rate of 4%, compounded monthly.

This is a future value of an ordinary annuity problem, where regular monthly deposits grow with interest over time. The formula to calculate the future value $FV$ of an ordinary annuity is:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value (in this case, $200,000).
• $P$ is the monthly deposit (what we're solving for).
• $r$ is the monthly interest rate.
• $n$ is the total number of deposits (months).

Step 1: Break down the known values

• Future value ($FV$) = $200,000
• Annual interest rate = 4% → Monthly interest rate ($r$) = $\frac{4\%}{12} = 0.003333$
• Total number of months ($n$) = 12 years × 12 months/year = 144 months

Step 2: Use the formula to find $P$

Rearrange the formula to solve for $P$:

$P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}$

Substitute the values:

$P = \frac{200,000}{\frac{(1 + 0.003333)^{144} - 1}{0.003333}}$

Step 3: Calculate

1. Calculate $(1 + 0.003333)^{144}$:

$(1 + 0.003333)^{144} \approx 1.60844$

2. Subtract 1:

$1.60844 - 1 = 0.60844$

3. Divide by $r = 0.003333$:

$\frac{0.60844}{0.003333} \approx 182.53$

4. Finally, divide $FV = 200,000$ by this value:

$P = \frac{200,000}{182.53} \approx 1,095.92$

Conclusion:

Emilia and Liam need to deposit $1,095.92 each month in order to accumulate $200,000 in 12 years at 4% interest, compounded monthly.

Would you like more details or have any questions?

Here are 5 related questions:

1. What if the interest rate were compounded annually instead of monthly?
2. How would the required monthly deposit change if the interest rate increased to 5%?
3. How much would they save after 10 years if they kept making the same monthly deposit?
4. What would happen if they increased their monthly deposit by 10% after 5 years?
5. What is the total amount of interest earned over the 12 years?

Tip: When dealing with annuities or loans, always check how often the interest is compounded—it significantly affects the total savings or payments!