three-overlapping-circlesIf a problem involves three overlapping sets, use a Venn diagram. This is simply the easiest method to find the solution. When you use Venn diagrams, always work from the inside out.

Example

There are three fitness clubs in Pasadena: Xerofit, Stayoung, and Zippy. Xerofit has 20 members, Stayoung has 30 members, and Zippy has 12 members. No one is in all three clubs but 8 are in both Xerofit and Stayoung, 6 are in both Xerofit and Zippy, and 4 are in both Stayoung and Zippy. How many people belong to at least one gym?

  1. 22
  2. 33
  3. 44
  4. 55
  5. 66

Solution

  1. The question is asking how many people belong to at least one gym.
  2. There are no variables in this problem. Organize the information provided into a Venn diagram. Start with how many people belong to all 3 clubs: 0. Then fill in the number of people that belong to 2 clubs.
  3. 3overlappingsets-example1

  4. The only mathematical relationship you need to understand is that each person can be counted only once regardless of how many clubs they belong to. eg. Xerofit has 20 members but 14 belong to another club as well. That means 6 members belong only to Xerofit. (20 – 8 – 6 = 6). Fill in the remaining circles
  5. 3overlappingsets-example2

  6. Solve. If we sum all of the numbers, we get 44. There are 44 people in at least one gym.



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