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Difficulty: MediumMust Be True Questions

According to the human resources records at Apex Corporation, all employees who participate in the profit-sharing plan have a minimum of five years of tenure. Furthermore, most employees with a minimum of five years of tenure belong to the labor union. Corporate policy dictates that no one who belongs to the labor union is eligible for a managerial bonus. Finally, records show that some employees who belong to the labor union also participate in the profit-sharing plan.

If the statements above are true, which one of the following must also be true?

  1. At least some employees who participate in the profit-sharing plan are ineligible for a managerial bonus.Answer
  2. B
    Most employees who participate in the profit-sharing plan belong to the labor union.
  3. C
    Anyone who has a minimum of five years of tenure participates in the profit-sharing plan.
  4. D
    If an employee is ineligible for a managerial bonus, that employee must belong to the labor union.
  5. E
    Employees who do not participate in the profit-sharing plan are ineligible for a managerial bonus.

Answer

At least some employees who participate in the profit-sharing plan are ineligible for a managerial bonus.
The stimulus establishes that some labor union members participate in the profit-sharing plan. It also establishes a universal rule that absolutely no labor union members are eligible for a managerial bonus. Combining these two facts means that the specific employees who are in both the union and the profit-sharing plan cannot be eligible for a bonus. Therefore, there must be at least some profit-sharing participants who are ineligible for a managerial bonus.

Step-by-Step Solution

1
Map the universal and majority rules provided in the stimulus.
Profit-sharing participants (PSP) -> 5+ years tenure (5Y). 5+ years tenure (5Y) -most-> Labor union (LU). Labor union (LU) -> No managerial bonus (~MB).
Creating a formal logic map of the premises ensures accurate deductions and prevents invalid leaps.
2
Map the existential (some) rule provided.
Labor union (LU) -some-> Profit-sharing participants (PSP).
Quantifier statements ('some', 'most') interact with universal statements ('all', 'none') in very specific ways on the exam.
3
Combine the 'some' statement with the relevant universal rule.
Because LU -some-> PSP, there is an overlapping group of people who are both LU and PSP. Because LU -> ~MB, every single person in the LU group is ~MB.
If you have a 'some' relationship (A some B) and a universal relationship for one of those terms (A all C), you can deduce a new 'some' relationship (B some C).
4
Draw the final deduction.
Since those overlapping employees are in the union, they cannot get a bonus. Therefore, since they are also in the profit-sharing plan, some profit-sharing participants cannot get a bonus (PSP -some-> ~MB).
This matches the correct answer perfectly.

Key Concept

Combining existential quantifiers ('some') with universal conditionals ('all'/'none') to form deductions.
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