Questions: You deposit 400 each month into an account earning 7% interest compounded monthly. a) How much will you have in the account in 25 years? b) How much total money will you put into the account? c) How much total interest will you earn?

Solution Steps

To solve this problem, we need to use the formula for the future value of a series of monthly deposits into an account with compound interest. The formula is:

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

where:

• \( P \) is the monthly deposit
• \( r \) is the monthly interest rate
• \( n \) is the total number of deposits

For part (b), we simply multiply the monthly deposit by the total number of months. For part (c), we subtract the total deposits from the future value to find the total interest earned.

To find the amount in the account after 25 years with monthly deposits of \( \$400 \) and an annual interest rate of \( 7\% \) compounded monthly, we use the future value formula:

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

where:

• \( P = 400 \)
• \( r = \frac{0.07}{12} = 0.0058333333 \)
• \( n = 25 \times 12 = 300 \)

Substituting the values, we get:

\[ FV = 400 \times \frac{(1 + 0.0058333333)^{300} - 1}{0.0058333333} \approx 324028.6772 \]

The total amount deposited over 25 years is calculated as:

\[ \text{Total Deposits} = P \times n = 400 \times 300 = 120000 \]

The total interest earned is the difference between the future value and the total deposits:

\[ \text{Total Interest} = FV - \text{Total Deposits} \approx 324028.6772 - 120000 = 204028.6772 \]

Final Answer