Question: User: a. Suppose that between the ages of $22$ and $32$, you contribute $9000$ per year to a $401(k)$ and your employer contributes $4500$ per year on your behalf. The interest rate is $7.9\%$ compounded annually. What is the value of the $401(k)$ after $10$ years? b. Suppose that after $10$ years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the $401(k)$. How much money will you have in the plan when you reach age $65$? c. What is the difference between the amount of money you will have accumulated in the $401(k)$ and the amount you contributed to the plan?

1. **State the problem:** We have annual contributions to a $401(k)$ plan from age $22$ to $32$ (10 years), with $9000$ from you and $4500$ from your employer each year. The interest rate is $7.9\%$ compounded annually. We want to find: a. The value of the $401(k)$ after $10$ years. b. The value of the $401(k)$ at age $65$ if no more contributions are made. c. The difference between the accumulated amount and the total contributions. 2. **Formula for future value of an annuity with annual contributions:** $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $P$ is the annual contribution - $r$ is the annual interest rate (as a decimal) - $n$ is the number of years contributions are made 3. **Calculate total annual contribution:** $$P = 9000 + 4500 = 13500$$ 4. **Calculate the future value after 10 years of contributions:** Given $r = 0.079$, $n = 10$, $$FV = 13500 \times \frac{(1 + 0.079)^{10} - 1}{0.079}$$ Calculate $(1 + 0.079)^{10}$: $$1.079^{10} \approx 2.129$$ So, $$FV = 13500 \times \frac{2.129 - 1}{0.079} = 13500 \times \frac{1.129}{0.079}$$ Calculate the fraction: $$\frac{1.129}{0.079} \approx 14.291$$ Therefore, $$FV = 13500 \times 14.291 = 192,878.5$$ So, the value after 10 years is approximately $192,878.50$. 5. **Calculate the value at age 65 (after 33 more years) with no additional contributions:** The amount will grow by compound interest for $33$ years: $$FV_{65} = FV \times (1 + r)^{33} = 192,878.5 \times 1.079^{33}$$ Calculate $1.079^{33}$: $$1.079^{33} \approx 10.02$$ So, $$FV_{65} = 192,878.5 \times 10.02 = 1,932,644.3$$ The value at age 65 is approximately $1,932,644.30$. 6. **Calculate total contributions:** You contributed $9000$ per year for $10$ years: $$9000 \times 10 = 90,000$$ Employer contributed $4500$ per year for $10$ years: $$4500 \times 10 = 45,000$$ Total contributions: $$90,000 + 45,000 = 135,000$$ 7. **Calculate the difference between accumulated amount at age 65 and total contributions:** $$1,932,644.3 - 135,000 = 1,797,644.3$$ **Final answers:** - a. Value after 10 years: approximately $192,878.50$ - b. Value at age 65: approximately $1,932,644.30$ - c. Difference: approximately $1,797,644.30$