Final answer:
The mathematics question compares retirement savings strategies using compound interest and annuities. The answer includes calculations to demonstrate how starting early with contributions can result in a greater amount at retirement due to the power of compound interest.
Step-by-step explanation:
The scenario presents a comparison of two different strategies for saving for retirement, highlighting the impact of compound interest over time. One individual contributes $2,000 annually to a retirement account for nine years and then ceases payments at the age of 45.
In contrast, their twin starts contributing $2,000 annually at the age 45 and continues until age 65, over a span of 20 years. In both cases, an annual interest rate of 9% is earned. To determine how much each will have at retirement, we'll need to calculate the future value of an annuity for both strategies.
For the individual who starts saving earlier, we use the formula for the future value of an annuity:
$2,000 × ¶(1 + 0.09) ¹ ¹ - 1)/0.09)
For the twin who starts later, we use the same formula:
$2,000 × ((1 + 0.09) ² ° - 1)/0.09)
Once we have both future values, we can compare the amounts to understand the advantage of starting early and benefiting from compound interest over a longer period. The initial individual will have a head start in accumulation due to the additional compounding periods, despite contributing for fewer years overall.
As observed in other examples, even though Yelberton will earn a salary over 30 years and plans to save for retirement, it's the early savings that significantly augment his retirement funds due to the compound interest effect.