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Difficulty: EasySet Theory Concepts and Venn Diagrams

A health club surveyed 100100 members regarding their use of cardio machines and weight equipment. Of the members surveyed, 6060 use cardio machines, 5050 use weight equipment, and 1515 use neither cardio machines nor weight equipment. Which of the following statements must be true? Indicate all such statements.

  1. Exactly 2525 members use both cardio machines and weight equipment.Answer
  2. Exactly 3535 members use cardio machines but do not use weight equipment.Answer
  3. C
    Exactly 1010 members use both cardio machines and weight equipment.
  4. D
    Exactly 110110 members use at least one of the two types of equipment.
  5. Exactly 8585 members use at least one of the two types of equipment.Answer

Answer

The true statements are those stating that exactly 25 members use both types of equipment, exactly 35 members use cardio machines only, and exactly 85 members use at least one type of equipment.
The total surveyed population of 100100 consists of members using at least one type of equipment and members using neither (1515). Thus, 100−15=85100 - 15 = 85 members use at least one type of equipment. Using the inclusion-exclusion formula Union=Cardio+Weights−Intersection\text{Union} = \text{Cardio} + \text{Weights} - \text{Intersection}, we have 85=60+50−Intersection85 = 60 + 50 - \text{Intersection}, giving an intersection of 2525 members using both types of equipment. Subtracting 2525 from the 6060 total cardio users shows that 3535 members use cardio machines exclusively.

Step-by-Step Solution

1
Calculate the total number of members using at least one type of equipment.
Total using at least one type =100−15=85= 100 - 15 = 85.
Members using at least one type of equipment represent the total surveyed population minus those who use neither.
2
Calculate the number of members using both cardio machines and weight equipment using the principle of inclusion-exclusion.
Both=Cardio+Weights−At least one=60+50−85=25\text{Both} = \text{Cardio} + \text{Weights} - \text{At least one} = 60 + 50 - 85 = 25.
Adding the individual set counts double-counts the overlapping region, so subtracting the total union gives the intersection size.
3
Calculate the number of members using cardio machines only.
Cardio Only=Cardio−Both=60−25=35\text{Cardio Only} = \text{Cardio} - \text{Both} = 60 - 25 = 35.
Subtracting the intersection from the total cardio group yields the exclusive cardio users.

Key Concept

Principle of Inclusion-Exclusion for Two Sets
Estimated Time:1m 15s
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Topics

Set Theory Concepts and Venn Diagrams

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