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Question

Difficulty: EasyOverlapping Sets and Venn Diagrams

In a survey of 120120 corporate executives, 7070 executives read Magazine A, 5050 read Magazine B, and 2525 read both Magazine A and Magazine B. How many of the executives surveyed read neither Magazine A nor Magazine B?

  1. A
    0
  2. 25Answer
  3. C
    45
  4. D
    50
  5. E
    95

Answer

25 executives surveyed read neither Magazine A nor Magazine B.
By the inclusion-exclusion principle, the total number of executives reading at least one magazine is given by Set A+Set B−Both=70+50−25=95\text{Set A} + \text{Set B} - \text{Both} = 70 + 50 - 25 = 95. Since 120120 executives were surveyed in total, the number of executives who read neither magazine is 120−95=25120 - 95 = 25.

Step-by-Step Solution

1
Apply the Inclusion-Exclusion Principle for two overlapping sets to find the number of executives reading at least one magazine.
Total(At least one)=Magazine A+Magazine B−Both=70+50−25=95\text{Total(At least one)} = \text{Magazine A} + \text{Magazine B} - \text{Both} = 70 + 50 - 25 = 95.
Directly adding the counts for Magazine A and Magazine B counts executives who read both twice, so the intersection must be subtracted once.
2
Subtract the number of executives who read at least one magazine from the total number of executives surveyed.
Neither=Total Surveyed−Total(At least one)=120−95=25\text{Neither} = \text{Total Surveyed} - \text{Total(At least one)} = 120 - 95 = 25.
The universe of surveyed executives consists of those who read at least one magazine and those who read neither.

Key Concept

Overlapping Sets (Double-Matrix / Two-Set Venn Diagram)
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Topics

Overlapping Sets and Venn Diagrams

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