To solve this problem, we’ll use the future value of an ordinary annuity formula, given that interest is compounded monthly:

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

where:

• $FV$ is the future value you want to achieve, which is $300,000,
• $P$ is the monthly deposit (which we need to find),
• $r$ is the monthly interest rate,
• $n$ is the total number of deposits.

Let's go through each part.

Part (a): Monthly Deposit Calculation

1. Convert the annual interest rate to a monthly rate: $r = \frac{10\%}{12} = 0.10 / 12 = 0.008333\ldots$

2. Determine the total number of deposits (months): $n = 25 \times 12 = 300$

3. Rearrange the formula to solve for $P$: $P = \frac{FV \cdot r}{(1 + r)^n - 1}$

4. Plug in the values: $P = \frac{300,000 \cdot 0.008333}{(1 + 0.008333)^{300} - 1}$

Calculating this will yield the monthly deposit required.

Solution:

$P \approx 334.57$

So, you would need to deposit approximately $334.57 each month to reach $300,000 in 25 years.

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Part (b): Interest Earned

To find the interest earned, we’ll calculate the total amount deposited over time and subtract it from the future value.

1. Total contributions: $\text{Total Deposits} = P \times n = 334.57 \times 300 \approx 100,371$

2. Total interest earned: $\text{Interest Earned} = FV - \text{Total Deposits} = 300,000 - 100,371 \approx 199,629$

So, the total interest earned would be approximately $199,629.

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Let me know if you want more details on any part, or have any questions!

Here are five related questions to consider:

1. What if the interest were compounded annually instead of monthly?
2. How much more would you need if the goal were $500,000 instead of $300,000?
3. What would the monthly deposit be if the interest rate were 8% instead of 10%?
4. How does increasing the deposit frequency (e.g., bi-weekly deposits) impact the required monthly deposit?
5. What if the deposit period were only 20 years instead of 25?

Tip: For retirement savings, compounding frequency significantly affects the future value, so monthly compounding grows the amount faster than annual compounding!