This problem can be solved using the formula for the future value of an ordinary annuity, where the payments are made at regular intervals, and the interest is compounded at a certain rate. However, since the deposits are made at the beginning of each period, the formula for the future value of an annuity due is more appropriate:

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

Where:

• $P = 125$: Monthly deposit
• $r = \frac{10.2\%}{12} = \frac{0.102}{12} \approx 0.0085$: Monthly interest rate
• $n = (55 - 19) \cdot 12 = 36 \cdot 12 = 432$: Total number of months
• $FV$: Future value

Step 1: Calculate the Future Value

$FV = 125 \cdot \frac{(1 + 0.0085)^{432} - 1}{0.0085} \cdot (1 + 0.0085)$

Let's calculate this: Parameters P = 125 # monthly deposit annual_rate = 0.102 # annual interest rate r = annual_rate / 12 # monthly interest rate years = 55 - 19 # years from 19 to 55 n = years * 12 # total number of months

Future Value of an Annuity Due formula

FV = P * ((1 + r)**n - 1) / r * (1 + r) FVThe future value of the retirement fund at age 55 is approximately $559,493.02.

Step 2: Calculate the Total Deposits

The total deposits are the product of the monthly deposit and the total number of months: $\text{Total Deposits} = P \cdot n = 125 \cdot 432 = 54,000$

Step 3: Calculate the Interest Earned

The interest earned is the difference between the future value and the total deposits: $\text{Interest Earned} = FV - \text{Total Deposits} = 559,493.02 - 54,000$

Let's compute the interest: Total deposits and interest earned total_deposits = P * n interest_earned = FV - total_deposits total_deposits, interest_earnedThe total amount Rachael deposited is $54,000, and the interest earned is approximately $505,493.02.

Final Answer:

• Future Value (Total Amount at Age 55): $559,493.02
• Total Interest Earned: $505,493.02

Would you like me to show the step-by-step calculations or clarify any part of the solution?

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Related Questions:

1. What happens to the future value if the interest rate changes to 8%?
2. How much would Rachael save if she started at age 25 instead of 19?
3. What is the effect of doubling the monthly contribution on the future value?
4. How long would it take for Rachael to accumulate $1,000,000 at this rate?
5. What if Rachael deposited at the end of each month instead?

Tip:

When solving financial problems, ensure you understand whether the payments are made at the beginning or end of each period—it affects the future value significantly.