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Difficulty: EasyOverlapping Sets, Statistics, and Data Distributions

A class of 50 students took tests in both Mathematics and Science. How many students passed both tests?

(1) 35 students passed Mathematics and 30 students passed Science.
(2) 10 students failed both tests.

  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 correct option is the choice stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) leaves the number of students failing both tests unknown, while Statement (2) leaves the individual subject pass numbers unknown. Combined, the standard set equation Total=Group A+Group B−Both+Neither\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither} yields a unique value of 2525 for students passing both tests.

Step-by-Step Solution

1
Set up the overlapping sets formula for two groups
Total=Math+Science−Both+Neither\text{Total} = \text{Math} + \text{Science} - \text{Both} + \text{Neither}, which becomes 50=Math+Science−Both+Neither50 = \text{Math} + \text{Science} - \text{Both} + \text{Neither}.
This formula connects all four components of a two-group overlapping set.
2
Evaluate Statement (1) independently
Substituting Math=35\text{Math} = 35 and Science=30\text{Science} = 30 into the formula yields 50=35+30−Both+Neither50 = 35 + 30 - \text{Both} + \text{Neither}, or Both−Neither=15\text{Both} - \text{Neither} = 15.
Since Neither\text{Neither} is unknown, Both\text{Both} cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently
Substituting Neither=10\text{Neither} = 10 yields 50=Math+Science−Both+1050 = \text{Math} + \text{Science} - \text{Both} + 10.
Since Math\text{Math} and Science\text{Science} are unknown, Both\text{Both} cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
Combining all given values gives 50=35+30−Both+10⇒50=75−Both⇒Both=2550 = 35 + 30 - \text{Both} + 10 \Rightarrow 50 = 75 - \text{Both} \Rightarrow \text{Both} = 25.
A single unique value of 25 is obtained for the number of students who passed both tests.

Key Concept

Overlapping Sets (Two-Group Venn Diagram Formula)
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Topics

Overlapping Sets, Statistics, and Data Distributions

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