To calculate the total amount saved with regular monthly contributions, we can use the Future Value of a series formula for compound interest. The formula for the future value of an annuity is:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) is the future value of the annuity (total amount accumulated),
- \( P \) is the amount of each payment (monthly investment),
- \( r \) is the monthly interest rate (annual interest rate divided by 12),
- \( n \) is the total number of payments (number of months).
Given:
- Monthly investment \( P = 25 \) dollars,
- Annual interest rate \( 3.25% \) or \( 0.0325 \),
- Monthly interest rate \( r = \frac{0.0325}{12} \approx 0.00270833 \),
- Time period \( 30 \) years or \( 30 \times 12 = 360 \) months.
Now we can substitute the values into the formula:
\[ FV = 25 \times \frac{(1 + 0.00270833)^{360} - 1}{0.00270833} \]
First, calculate \( (1 + r)^{n} \):
\[ (1 + 0.00270833)^{360} \approx 2.619121 \]
Now substitute this value back into the formula:
\[ FV = 25 \times \frac{2.619121 - 1}{0.00270833} \]
Calculate \( 2.619121 - 1 = 1.619121 \) and then divide:
\[ \frac{1.619121}{0.00270833} \approx 597.3367 \]
Now multiply by \( 25 \):
\[ FV \approx 25 \times 597.3367 \approx 14933.42 \]
So, the future value of the investment after 30 years is approximately \( 14933.42 \) dollars.
Next, to calculate the total amount contributed, we find:
\[ \text{Total Contributions} = P \times n = 25 \times 360 = 9000 \text{ dollars} \]
Finally, to find the total interest earned, we subtract the total contributions from the future value:
\[ \text{Total Interest Earned} = FV - \text{Total Contributions} = 14933.42 - 9000 \approx 5933.42 \]
Thus, the total interest earned is approximately $5933.42.
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