Challenge question. In the chapter text, we dealt exclusively with a single lump sum, but often we may be looking at several lump-sum values simultaneously. Let's consider the retirement plan of a couple. Currently, the couple has four different investments: a \( 401(\mathrm{k}) \) plan, two pension plans, and a personal portfolio. The couple is 6 years away from retirement. They believe they have sufficient money in their plans today so that they do not have to contribute to the plans over the next 6 years and will still meet their \( \$ 1.5 \) million retirement goal. Here are the current values and growth rates of their plans: \( 401(\mathrm{k}): \$ 93,000 \) growing at \( 6 \% \). Pension Plan One: \( \$ 287,000 \) growing at \( 6.75 \% \). Pension Plan Two: \( \$ 250,000 \) growing at \( 7.5 \% \). Personal Portfolio: \( \$ 141,000 \) growing at \( 10 \% \). What is the value of the \( 401(\mathrm{k}) 6 \) years from now?

To find the value of the \(401(k)\) plan 6 years from now, we can use the formula for compound interest: \[ A = P(1 + r)^n \] where: - \( A \) is the future value of the investment, - \( P \) is the present value of the investment, - \( r \) is the annual growth rate of the investment, - \( n \) is the number of years the investment will be growing. Given: - Present value of the \(401(k)\) plan: \( P = \$93,000 \) - Annual growth rate of the \(401(k)\) plan: \( r = 6\% = 0.06 \) - Number of years the investment will be growing: \( n = 6 \) Substitute these values into the formula to find the future value of the \(401(k)\) plan 6 years from now. Calculate the value by following steps: - step0: Calculate: \(93000\left(1+0.06\right)^{6}\) - step1: Add the numbers: \(93000\times 1.06^{6}\) - step2: Convert the expressions: \(93000\left(\frac{53}{50}\right)^{6}\) - step3: Simplify: \(93000\times \frac{53^{6}}{50^{6}}\) - step4: Rewrite the expression: \(50\times 1860\times \frac{53^{6}}{50^{6}}\) - step5: Reduce the numbers: \(1860\times \frac{53^{6}}{50^{5}}\) - step6: Rewrite the expression: \(10\times 186\times \frac{53^{6}}{50^{5}}\) - step7: Rewrite the expression: \(10\times 186\times \frac{53^{6}}{10^{5}\times 5^{5}}\) - step8: Reduce the numbers: \(186\times \frac{53^{6}}{10^{4}\times 5^{5}}\) - step9: Rewrite the expression: \(2\times 93\times \frac{53^{6}}{10^{4}\times 5^{5}}\) - step10: Rewrite the expression: \(2\times 93\times \frac{53^{6}}{2^{4}\times 5^{4}\times 5^{5}}\) - step11: Reduce the numbers: \(93\times \frac{53^{6}}{2^{3}\times 5^{4}\times 5^{5}}\) - step12: Multiply: \(\frac{93\times 53^{6}}{15625000}\) The value of the \(401(k)\) plan 6 years from now is approximately $131,922.28.