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Difficulty: MediumOverlapping Sets, Statistics, and Data Distributions

In a technology consulting firm of 100 employees, every employee works in either the Analytics department, the Engineering department, or both. How many employees work in both departments?

(1) Exactly 70 employees work in the Analytics department, and 60 employees work in the Engineering department.
(2) Exactly 40 employees work ONLY in the Analytics department.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct answer states that Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. Using the standard formula for overlapping sets, Total = Group A + Group B - Both, Statement (1) provides values for Total (100), Group A (70), and Group B (60), allowing us to solve directly for Both = 30. Statement (2) only specifies the number of members belonging strictly to Group A, which leaves the overlap dependent on the unstated size of Group B.

Step-by-Step Solution

1
Set up the overlapping sets formula for two groups.
Total = N(Analytics) + N(Engineering) - N(Both)
Since every employee belongs to at least one of the two departments, the union of the two sets equals the total number of employees, 100.
2
Evaluate Statement (1) independently.
100 = 70 + 60 - N(Both) => N(Both) = 30.
Statement (1) provides N(Analytics) = 70 and N(Engineering) = 60. Substituting these values into the formula yields a unique value of 30 for N(Both). Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently.
100 = N(Only Analytics) + N(Only Engineering) + N(Both) => 100 = 40 + N(Only Engineering) + N(Both).
We have two unknown variables: N(Only Engineering) and N(Both). Multiple non-negative integer pairs satisfy this equation (e.g., N(Both) could be 0, 10, 20, etc.). Thus, Statement (2) alone is not sufficient.

Key Concept

Overlapping Sets Formula for Two Groups
Estimated Time:1m 30s
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Topics

Overlapping Sets, Statistics, and Data Distributions

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