Emilia and Liam are purchasing a home. They wish to save money for 12
 years and purchase a house that has a value of $200,000
 with cash. If they deposit money into an account paying 4%
 interest, compounded monthly, how much do they need to deposit each month in order to make the purchase? Round your answer to the nearest cent, if necessary.

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!

Emilia and Liam are purchasing a home. They wish to save money for 12 years and purchase a house that has a value of $200,000 with cash. If they deposit money into an account paying 4% interest, compounded monthly, how much do they need to deposit each month in order to make the purchase?

Find out how much Emilia and Liam need to deposit each month in an account paying 4% interest, compounded monthly, to save $200,000 over 12 years. This step-by-step solution covers key concepts in compound interest and the future value of an annuity, with a detailed breakdown of calculations.

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