Questions: Suppose you want to have 300,000 for retirement in 25 years. Your account earns 10% interest. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

Solution Steps

To solve this problem, we need to use the future value of an annuity formula to determine the monthly deposit required to reach a future value of $300,000 in 25 years with a 10% annual interest rate compounded monthly. Then, we can calculate the total amount deposited and subtract it from the future value to find the interest earned.

To find the monthly deposit \( D \) required to accumulate a future value \( FV \) of \$300,000 in 25 years at an annual interest rate of 10%, we use the future value of an annuity formula:

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

where:

• \( FV = 300,000 \)
• \( r = \frac{0.10}{12} = 0.0083333 \) (monthly interest rate)
• \( n = 25 \times 12 = 300 \) (total number of months)

Substituting the values, we find:

\[ D = \frac{300,000 \cdot 0.0083333}{(1 + 0.0083333)^{300} - 1} \approx 226.1022 \]

The total amount deposited over the 25 years is given by:

\[ \text{Total Deposited} = D \cdot n = 226.1022 \cdot 300 \approx 67,830.6710 \]

The interest earned can be calculated by subtracting the total amount deposited from the future value:

\[ \text{Interest Earned} = FV - \text{Total Deposited} = 300,000 - 67,830.6710 \approx 232,169.3290 \]

Final Answer