Final answer:
To find Jamal's retirement age, calculate the growth of his retirement funds with compound interest and annual contributions until it's sufficient to provide $100,000 annually for 21 years. Using the future value formula for an investment with contributions, determine when the fund reaches the required amount for Jamal to retire.
Step-by-step explanation:
To determine the youngest age at which Jamal can retire, we need to calculate how the accumulated and annual additions to his retirement fund will grow over time using compound interest, and at what age these funds would be enough to provide an annual income of $100,000 until he is projected to live to 76 years old.
According to the premise, Jamal has accumulated $200,000 and is adding $10,000 per year. The fund earns 5% interest. Jamal requires an annual income of $100,000 from his retirement fund from the time he retires until he reaches age 76, meaning he needs enough in the fund to cover 21 years of retirement income without any additional contributions.
First, we need to determine the future value of Jamal's retirement fund each year. We will use the future value formula for an investment with annual contributions: Future Value = Present Value × (1 + interest rate)^number of periods + Annual Contribution × [((1 + interest rate)^number of periods - 1) / interest rate]. Here, Jamal's Present Value is $200,000, Annual Contribution is $10,000, and interest rate is 5% (or 0.05).
The life expectancy table indicates that Jamal is expected to live for another 21 years. Therefore, he needs to have accumulated enough to withdraw $100,000 annually for 21 years, without depleting the fund before. We can solve this through trial and error, calculating the future value each year until Jamal's fund reaches the required amount to retire.
Finally, once we have determined the year in which Jamal's fund reaches the target amount to generate an annual income of $100,000 for 21 years, we will know the youngest age at which he can retire.