This is a future value of an annuity problem where we want to determine the monthly deposit that will accumulate to $300,000 over 25 years with a 6% annual interest rate, compounded monthly.

Part (a): How much to deposit each month?

We can use the future value of an annuity formula:

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

Where:

• $FV$ is the future value ($300,000),
• $P$ is the monthly deposit (which we are solving for),
• $r$ is the monthly interest rate (annual rate divided by 12),
• $n$ is the total number of payments (number of months).

First, calculate the parameters:

• Annual interest rate = 6%, so monthly interest rate $r = \frac{6\%}{12} = 0.005$,
• Time in years = 25, so number of months $n = 25 \times 12 = 300$.

Now, rearranging the formula to solve for $P$ (the monthly deposit):

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

Substitute the known values:

$P = \frac{300,000 \times 0.005}{(1 + 0.005)^{300} - 1}$

Let's compute this.

The monthly deposit you would need to make is approximately $432.90.

Part (b): How much interest will you earn?

The total amount you will contribute over 25 years is:

$\text{Total Contributions} = P \times n$

The interest earned will be:

$\text{Interest Earned} = FV - \text{Total Contributions}$

Let's calculate this now.The total contributions over 25 years would be approximately $129,871.26, and the interest earned would be approximately $170,128.74.

Would you like any further details or clarification?

Here are 5 related questions to explore:

1. What if the interest rate was 5% instead of 6%?
2. How would the total contributions and interest change with a 20-year retirement plan?
3. What is the effect of contributing a larger amount initially versus regular monthly contributions?
4. How would switching from monthly to quarterly contributions affect the plan?
5. How much will be saved if the interest is compounded annually instead of monthly?

Tip: The longer you allow your money to grow in a retirement account, the more interest it will accrue due to compounding effects.