Questions: Annette Michaelson will need 19,000 in 8 years to help pay for her education. Determine the lump sum, deposited today at 3.5% compounded monthly, will produce the necessary amount.

Solution Steps

To determine the lump sum that needs to be deposited today to accumulate to $19,000 in 8 years with an interest rate of 3.5% compounded monthly, we use the formula for compound interest: \( A = P \left(1 + \frac{r}{m}\right)^{mt} \). Here, \( A \) is the future value ($19,000), \( P \) is the present value (the lump sum we need to find), \( r \) is the annual interest rate (0.035), \( m \) is the number of compounding periods per year (12), and \( t \) is the number of years (8). We solve for \( P \).

We are given the following values:

• Future value \( A = 19000 \)
• Annual interest rate \( r = 0.035 \)
• Compounding periods per year \( m = 12 \)
• Number of years \( t = 8 \)

We use the formula for compound interest to find the present value \( P \): \[ A = P \left(1 + \frac{r}{m}\right)^{mt} \] Rearranging the formula to solve for \( P \): \[ P = \frac{A}{\left(1 + \frac{r}{m}\right)^{mt}} \]

Substituting the known values into the equation: \[ P = \frac{19000}{\left(1 + \frac{0.035}{12}\right)^{12 \times 8}} \]

Calculating the expression: \[ P = \frac{19000}{\left(1 + 0.00291667\right)^{96}} \approx \frac{19000}{\left(1.00291667\right)^{96}} \approx \frac{19000}{1.348850} \] This results in: \[ P \approx 14365.7445 \]

Final Answer