Suppose you want to have $400,000 for retirement in 25 years. Your account earns 6% interest.

a) How much would you need to deposit in the account each month?

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b) How much interest will you earn?

To solve this problem, we will use the formula for the future value of an ordinary annuity, since you are making regular monthly deposits and earning interest over time. The formula for the future value $FV$ of an ordinary annuity is:

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

Where:
- $FV$ is the future value of the account (in this case, \$400,000),
- $P$ is the monthly deposit (what we are solving for),
- $r$ is the monthly interest rate,
- $n$ is the total number of deposits.

### Step-by-step breakdown:

- The annual interest rate is 6%, so the monthly interest rate $r$ is:
  \[
  r = \frac{6\%}{12} = 0.005
  \]

- The total number of months $n$ over 25 years is:
  \[
  n = 25 \times 12 = 300
  \]

#### a) Solving for $P$:
We are given $FV = 400,000$, and we need to solve for $P$ using the formula:
\[
400,000 = P \times \frac{(1 + 0.005)^{300} - 1}{0.005}
\]

Let's calculate $P$.The monthly deposit required is approximately \$577.21.

#### b) To find the total interest earned:
We first calculate the total amount of money deposited, which is:
\[
\text{Total Deposits} = P \times n
\]
Then, the interest earned is:
\[
\text{Interest Earned} = FV - \text{Total Deposits}
\]

Let's calculate the total interest earned.The total interest earned over the 25 years is approximately \$226,838.32.

### Summary:
a) You would need to deposit about **\$577.21** each month.
b) The total interest you will earn is approximately **\$226,838.32**.

Would you like more details or have any questions about this solution?

Here are some related questions:
1. What would the deposit amount be if the interest rate were 5%?
2. How much would you need to deposit if you want \$500,000 for retirement?
3. How does compound interest affect long-term savings?
4. What would happen if you made yearly deposits instead of monthly?
5. How much will you save with different deposit frequencies (e.g., weekly)?

**Tip:** Regular, consistent contributions, even if small, combined with compound interest, can significantly grow your savings over time.

Suppose you want to have $400,000 for retirement in 25 years. Your account earns 6% interest.

a) How much would you need to deposit in the account each month?

b) How much interest will you earn?

Learn how to calculate the monthly deposit needed to save $400,000 for retirement in 25 years with 6% interest. This problem explores compound interest and the future value of an ordinary annuity. Discover how much interest you'll earn over time. Suitable for high school and college students.

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