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Difficulty: HardOverlapping Sets and Venn Diagrams

A commercial airline conducted a fleet audit of 180180 transoceanic aircraft to evaluate their emergency navigation systems: Satellite Positioning (SS), Inertial Guidance (II), and Terrestrial Radio Navigation (TT). The audit revealed the following data:
- 9595 aircraft are equipped with Satellite Positioning.
- 8080 aircraft are equipped with Inertial Guidance.
- 7575 aircraft are equipped with Terrestrial Radio Navigation.
- 3030 aircraft are equipped with Satellite Positioning and Inertial Guidance, but not Terrestrial Radio Navigation.
- 2020 aircraft are equipped with Inertial Guidance and Terrestrial Radio Navigation, but not Satellite Positioning.
- 2525 aircraft are equipped with Satellite Positioning and Terrestrial Radio Navigation, but not Inertial Guidance.
- 1515 aircraft are not equipped with any of these three navigation systems.

How many aircraft in the fleet are equipped with exactly one of these three navigation systems?

  1. A
    7070
  2. B
    7575
  3. 8585Answer
  4. D
    9595
  5. E
    100100

Answer

The number of aircraft equipped with exactly one navigation system is 8585.
Subtracting the 1515 unequipped aircraft from the total fleet of 180180 gives 165165 aircraft equipped with at least one navigation system. Letting xx represent the number of aircraft with all three systems, the single-system counts are (95−30−25−x)=40−x(95 - 30 - 25 - x) = 40 - x for Satellite Positioning only, (80−30−20−x)=30−x(80 - 30 - 20 - x) = 30 - x for Inertial Guidance only, and (75−25−20−x)=30−x(75 - 25 - 20 - x) = 30 - x for Terrestrial Radio Navigation only. Summing all seven mutually exclusive regions yields (40−x)+(30−x)+(30−x)+30+20+25+x=165(40 - x) + (30 - x) + (30 - x) + 30 + 20 + 25 + x = 165, which simplifies to 175−2x=165175 - 2x = 165, giving x=5x = 5. Substituting x=5x = 5 into the single-system expressions gives 35+25+25=8535 + 25 + 25 = 85.

Step-by-Step Solution

1
Calculate the total number of aircraft equipped with at least one system.
180−15=165180 - 15 = 165 aircraft.
Subtracting the 1515 aircraft with no systems from the fleet total gives the union of sets SS, II, and TT.
2
Express the regions of the 3-set Venn diagram in terms of xx, where xx is the number of aircraft equipped with all three systems.
Only S=40−xS = 40 - x, Only I=30−xI = 30 - x, Only T=30−xT = 30 - x.
For set SS: 95−(30+25+x)=40−x95 - (30 + 25 + x) = 40 - x. For set II: 80−(30+20+x)=30−x80 - (30 + 20 + x) = 30 - x. For set TT: 75−(25+20+x)=30−x75 - (25 + 20 + x) = 30 - x.
3
Set up an equation for the total number of aircraft with at least one system to solve for xx.
(40−x)+(30−x)+(30−x)+30+20+25+x=165  ⟹  175−2x=165  ⟹  x=5(40 - x) + (30 - x) + (30 - x) + 30 + 20 + 25 + x = 165 \implies 175 - 2x = 165 \implies x = 5.
Summing all 77 non-overlapping regions inside the Venn diagram equals the total union of 165165.
4
Compute the sum of aircraft equipped with exactly one system.
(Only SS) + (Only II) + (Only TT) = (40−5)+(30−5)+(30−5)=35+25+25=85(40 - 5) + (30 - 5) + (30 - 5) = 35 + 25 + 25 = 85.
Substitute x=5x = 5 into each of the single-system region expressions.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Decomposition
Estimated Time:2m 0s
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Topics

Overlapping Sets and Venn Diagrams

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