Andrea, a self-employed individual, wishes to accumulate a retirement fund of $400,000. How much should she deposit each month into her retirement account, which pays interest at a rate of 3.5%/year compounded monthly, to reach her goal upon retirement 40 years from now? (Round your answer to the nearest cent.) 

To find out how much Andrea needs to deposit each month to accumulate a retirement fund of $400,000, we will use the formula for the future value of an ordinary annuity, since the deposits are made at the end of each month.

### Step-by-Step Solution

1. **Identify the known values**:
   - Future Value (FV) = $400,000
   - Annual interest rate = 3.5%
   - Interest rate per month ($r$) = $\frac{3.5\%}{12} = \frac{0.035}{12} \approx 0.0029167$
   - Number of years = 40
   - Number of compounding periods per year = 12 (monthly)
   - Total number of periods ($n$) = 40 years $\times$ 12 months/year = 480 months

2. **Formula for the Future Value of an Ordinary Annuity**:
   \[
   FV = P \times \frac{(1 + r)^n - 1}{r}
   \]
   where:
   - $FV$ is the future value,
   - $P$ is the monthly deposit,
   - $r$ is the monthly interest rate,
   - $n$ is the total number of deposits.

3. **Rearrange the formula to solve for $P$ (the monthly deposit)**:
   \[
   P = \frac{FV \times r}{(1 + r)^n - 1}
   \]

4. **Plug in the known values**:
   \[
   P = \frac{400,000 \times 0.0029167}{(1 + 0.0029167)^{480} - 1}
   \]

5. **Calculate the monthly deposit $P$**:

Let's compute this step-by-step:

\[
(1 + 0.0029167)^{480} \approx 3.920016
\]

\[
(1 + 0.0029167)^{480} - 1 \approx 3.920016 - 1 = 2.920016
\]

\[
P = \frac{400,000 \times 0.0029167}{2.920016}
\]

\[
P \approx \frac{1166.68}{2.920016} \approx 399.43
\]

Therefore, Andrea should deposit **$399.43** each month into her retirement account to reach her goal of $400,000 upon retirement in 40 years.

Would you like a further breakdown of these calculations, or do you have any other questions?

### Related Questions

1. What if Andrea wanted to accumulate a retirement fund of $500,000 instead? How would her monthly deposit change?
2. How much interest will Andrea have earned on her retirement fund over the 40 years?
3. If the interest rate was 4% instead of 3.5%, how would Andrea's monthly deposit change?
4. How does changing the compounding frequency (e.g., from monthly to quarterly) affect the amount Andrea needs to deposit?
5. What if Andrea wanted to retire in 30 years instead of 40 years? How would this affect her monthly deposit?

**Tip:** Consistently making contributions, especially with compounding interest, is crucial for long-term savings growth.

Learn how much Andrea needs to deposit each month into her retirement account to accumulate $400,000 with a 3.5% annual interest rate over 40 years. Detailed step-by-step solution using compound interest principles. Suitable for high school students and individuals interested in financial planning.

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