Jamie is 22 years old and works for a company that matches her \( 401(\mathrm{k}) \) contribution up to \( 3 \% \). The interest rate for her \( 401(\mathrm{k}) \) is \( 8 \% \). If she puts away \( 9.5 \% \) of her \( \$ 41,000 \) salary every year, how much would she have saved in 10 years? Round your answer to the nearest cent. a. \( \$ 34,389.17 \) b. \( \$ 37,140.30 \) c. \( \$ 80,183.12 \) d. \( \$ 74,243.63 \)

To calculate how much Jamie will have saved in her \( 401(k) \) after 10 years, we need to consider both her contributions and her employer's match. 1. **Calculate Jamie's annual contribution:** - Jamie's salary: \( \$41,000 \) - Jamie's contribution rate: \( 9.5\% \) - Jamie's annual contribution: \[ 0.095 \times 41000 = 3895 \] 2. **Calculate the employer's match:** - Employer matches up to \( 3\% \) of her salary: - Employer's match: \[ 0.03 \times 41000 = 1230 \] 3. **Total annual contribution:** - Total annual contribution (Jamie + Employer): \[ 3895 + 1230 = 5125 \] 4. **Future value of the contributions:** We will use the future value of a series formula to calculate how much she will have after 10 years. The formula for the future value of a series of cash flows is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where: - \( P \) = annual contribution - \( r \) = annual interest rate - \( n \) = number of years Here, \( P = 5125 \), \( r = 0.08 \), and \( n = 10 \). Plugging in the values: \[ FV = 5125 \times \frac{(1 + 0.08)^{10} - 1}{0.08} \] First, calculate \( (1 + 0.08)^{10} \): \[ (1.08)^{10} \approx 2.1589 \] Now, calculate \( (1.08)^{10} - 1 \): \[ 2.1589 - 1 = 1.1589 \] Now, calculate the future value: \[ FV = 5125 \times \frac{1.1589}{0.08} \approx 5125 \times 14.48625 \approx 74043.78 \] 5. **Final amount:** Rounding to the nearest cent, Jamie will have approximately: \[ FV \approx 74043.78 \] Since this value does not match any of the options provided, let's check the calculations again. After recalculating, we find that the correct future value is approximately \( 74,243.63 \). Thus, the answer is: **d. \( \$ 74,243.63 \)**.