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Difficulty: MediumLogical Venn Diagrams

A municipal transport authority conducted a survey of 500500 daily commuters to analyze their usage of three public transport modes: Bus, Metro, and Train. The survey revealed the following data:
- 210210 commuters use the Bus.
- 190190 commuters use the Metro.
- 160160 commuters use the Train.
- 7070 commuters use exactly two of these transport modes.
- 2020 commuters use all three transport modes.

How many commuters in the surveyed group use none of these three transport modes?

  1. A
    30
  2. 50Answer
  3. C
    70
  4. D
    140

Answer

The number of commuters using none of the three transport modes is 50.
The correct answer is found by determining the total number of commuters using at least one transport mode (the union). The sum of the individual sets (210+190+160=560210 + 190 + 160 = 560) counts people using exactly one mode once, exactly two modes twice, and exactly three modes three times. Therefore, 560=(Exactly One)+2×(Exactly Two)+3×(Exactly Three)560 = (\text{Exactly One}) + 2 \times (\text{Exactly Two}) + 3 \times (\text{Exactly Three}). Substituting the given values gives 560=(Exactly One)+2(70)+3(20)560 = (\text{Exactly One}) + 2(70) + 3(20), which simplifies to 560=(Exactly One)+140+60560 = (\text{Exactly One}) + 140 + 60. Thus, Exactly One equals 360360. The total union is Exactly One ++ Exactly Two ++ Exactly Three, which is 360+70+20=450360 + 70 + 20 = 450. Finally, the number of commuters using none of the modes is the total surveyed minus the union: 500−450=50500 - 450 = 50.

Step-by-Step Solution

1
Calculate the sum of commuters in the individual transport mode categories.
210+190+160=560210 + 190 + 160 = 560.
This establishes the gross total before adjusting for overlaps, where multi-mode commuters are counted multiple times.
2
Set up the inclusion-exclusion relationship for disjoint regions.
560=n(Exactly 1)+2×n(Exactly 2)+3×n(Exactly 3)560 = n(\text{Exactly 1}) + 2 \times n(\text{Exactly 2}) + 3 \times n(\text{Exactly 3}).
When summing the individual sets, commuters using exactly two modes are counted twice, and those using all three are counted three times.
3
Solve for the number of commuters using exactly one mode.
560=n(Exactly 1)+2(70)+3(20)⇒560=n(Exactly 1)+200⇒n(Exactly 1)=360560 = n(\text{Exactly 1}) + 2(70) + 3(20) \Rightarrow 560 = n(\text{Exactly 1}) + 200 \Rightarrow n(\text{Exactly 1}) = 360.
We need the 'exactly one' count to piece together the total union of commuters using at least one mode.
4
Calculate the total number of commuters using at least one mode (the union).
Union=360+70+20=450\text{Union} = 360 + 70 + 20 = 450 commuters.
The union is simply the sum of the disjoint regions: exactly one, exactly two, and exactly three.
5
Subtract the union from the total surveyed commuters to find those using none.
500−450=50500 - 450 = 50 commuters.
Commuters outside the union represent those who use zero of the surveyed transport modes.

Key Concept

Solving 3-set Venn diagrams using disjoint region equations rather than standard intersection formulas.
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Topics

Logical Venn Diagrams

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