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Difficulty: EasyOverlapping Sets, Statistics, and Data Distributions

A department has 4040 employees, and each employee speaks at least one of two languages: French or Spanish. How many employees in the department speak Spanish?

(1) 2525 employees speak French.
(2) 1010 employees speak both French and Spanish.

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
The total number of employees who speak at least one language is given by Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}. Neither statement alone provides both the number of French speakers and the number of employees who speak both languages. However, combining both statements gives 40=25+Spanish−1040 = 25 + \text{Spanish} - 10, which uniquely solves to Spanish=25\text{Spanish} = 25. Therefore, both statements together are sufficient, but neither alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem using the overlapping sets formula.
Since every employee speaks at least one language, Neither=0\text{Neither} = 0. The relationship is Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}, which simplifies to 40=French+Spanish−Both40 = \text{French} + \text{Spanish} - \text{Both}.
Establishing the mathematical relationship before evaluating statements clarifies what data is missing.
2
Evaluate Statement (1) independently.
Statement (1) gives French=25\text{French} = 25. Substituting this gives 40=25+Spanish−Both40 = 25 + \text{Spanish} - \text{Both}, or Spanish−Both=15\text{Spanish} - \text{Both} = 15.
Since Both\text{Both} is unknown, Spanish\text{Spanish} could take multiple values. Statement (1) is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives Both=10\text{Both} = 10. Substituting this gives 40=French+Spanish−1040 = \text{French} + \text{Spanish} - 10, or French+Spanish=50\text{French} + \text{Spanish} = 50.
Since French\text{French} is unknown, Spanish\text{Spanish} cannot be uniquely determined. Statement (2) is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
Substitute both values into the equation: 40=25+Spanish−10  ⟹  40=15+Spanish  ⟹  Spanish=2540 = 25 + \text{Spanish} - 10 \implies 40 = 15 + \text{Spanish} \implies \text{Spanish} = 25.
The equation yields a single, unique value for the target variable. Both statements together are sufficient.

Key Concept

Overlapping Sets (Two Groups)
Estimated Time:1m 0s
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Topics

Overlapping Sets, Statistics, and Data Distributions

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