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Difficulty: HardLogical Venn Diagrams

In a municipal department of 120120 officers, a survey was conducted regarding three operational responsibilities: Urban Planning (UU), Environmental Compliance (EE), and Public Works (PP). The survey revealed the following data:
- 6565 officers handle Urban Planning (UU).
- 5050 officers handle Environmental Compliance (EE).
- 5555 officers handle Public Works (PP).
- 55 officers handle none of these three responsibilities.
- 4545 officers handle at least two of these responsibilities.
- 1010 officers handle all three responsibilities.
- 2020 officers handle both Urban Planning and Environmental Compliance.
- 1818 officers handle both Environmental Compliance and Public Works.

How many officers handle ONLY Urban Planning?

  1. A
    18
  2. 28Answer
  3. C
    35
  4. D
    38

Answer

The number of officers handling ONLY Urban Planning is 28.
Subtracting all overlapping regions connected to Urban Planning (1010 for Urban Planning & Environmental only, 1717 for Urban Planning & Public Works only, and 1010 for all three) from the total Urban Planning count (6565) leaves 65−37=2865 - 37 = 28 officers handling Urban Planning exclusively.

Step-by-Step Solution

1
Determine the total number of officers handling at least one responsibility.
Total in at least one set = 120−5=115120 - 5 = 115.
5 officers handle none of the three tasks.
2
Identify the regions of overlapping sets using given intersection values.
Let the 3-set intersection be g=10g = 10. The region for ONLY Urban Planning and Environmental Compliance is d=20−10=10d = 20 - 10 = 10. The region for ONLY Environmental Compliance and Public Works is e=18−10=8e = 18 - 10 = 8.
The total count for two sets includes those who also handle all three.
3
Calculate the region for ONLY Urban Planning and Public Works (ff).
Since officers handling at least two responsibilities equals d+e+f+g=45d + e + f + g = 45, we have 10+8+f+10=45  ⟹  f=1710 + 8 + f + 10 = 45 \implies f = 17.
The sum of all two-set-only regions and the three-set region equals 45.
4
Compute the number of officers handling ONLY Urban Planning (aa).
a=n(U)−(d+f+g)=65−(10+17+10)=65−37=28a = n(U) - (d + f + g) = 65 - (10 + 17 + 10) = 65 - 37 = 28.
Subtracting all shared regions from the total Urban Planning count leaves the officers dedicated solely to Urban Planning.

Key Concept

3-Set Venn Diagram Region Decomposition using Inclusion-Exclusion Principle
Estimated Time:2m 30s
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Topics

Logical Venn Diagrams

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