Venn Diagram Problem: Runners, Bowlers & Skiers Explained!

by TextBrain Team 59 views

Hey guys! Let's dive into a classic set theory problem involving a Venn diagram. This problem is all about figuring out how many people participate in different activities, and how many don't participate at all. We'll break it down step-by-step, so you can understand exactly how to solve these types of questions. So, let's get started and unravel this interesting problem together!

Problem Statement

Okay, so here's the situation: In a group of 70 people, we have some athletes who enjoy different sports. Specifically:

  • 34 people practice running.
  • 25 people practice bowling.
  • 27 people practice skiing.

Now, things get a bit more interesting because some people do multiple activities:

  • 7 people practice both running and bowling.
  • 11 people practice both running and skiing.
  • 6 people practice both skiing and bowling.

And to top it off, we have some super-athletes:

  • 5 people practice all three activities (running, bowling, and skiing).

Our mission, should we choose to accept it, is twofold:

A. Construct and name a suitable Venn diagram to represent this data.

B. Determine how many people do not participate in any of these three activities.

Sounds like a fun challenge, right? Let's put on our thinking caps and get to it!

Constructing the Venn Diagram

Understanding Venn Diagrams

First things first, let's quickly recap what a Venn diagram is. A Venn diagram is a visual tool that uses overlapping circles to illustrate the relationships between different sets. In our case, each circle will represent a sport: running, bowling, and skiing. The overlapping areas represent the people who participate in more than one sport.

Drawing the Circles

To start, we draw three overlapping circles. Label them as follows:

  • Circle A: Running
  • Circle B: Bowling
  • Circle C: Skiing

Make sure the circles overlap because we know some people participate in multiple activities.

Filling in the Intersections

This is where the fun begins! We'll work from the innermost intersection outwards.

  1. All Three Activities: We know 5 people practice all three sports. So, we write "5" in the region where all three circles (A, B, and C) overlap. This is the heart of our Venn diagram, where the core athletes reside.

  2. Running and Bowling: 7 people practice running and bowling. But, we've already accounted for 5 of them in the center (those who do all three). So, we subtract 5 from 7, leaving us with 2 people who practice only running and bowling. Write "2" in the overlapping region between circles A and B, but outside the center intersection.

  3. Running and Skiing: 11 people practice running and skiing. Again, we subtract the 5 who do all three, leaving 6 people who practice only running and skiing. Write "6" in the overlapping region between circles A and C, but outside the center.

  4. Skiing and Bowling: 6 people practice skiing and bowling. Subtract the 5 who do all three, leaving 1 person who practices only skiing and bowling. Write "1" in the overlapping region between circles B and C, but outside the center.

Filling in the Individual Circles

Now, let's fill in the remaining portions of each circle, representing those who participate in only one sport.

  1. Running: 34 people practice running in total. We've already accounted for:

    • 5 who do all three sports
    • 2 who do running and bowling
    • 6 who do running and skiing

    So, we subtract 5 + 2 + 6 = 13 from 34, which leaves us with 21 people who practice only running. Write "21" in the remaining part of circle A.

  2. Bowling: 25 people practice bowling in total. We've accounted for:

    • 5 who do all three sports
    • 2 who do running and bowling
    • 1 who does skiing and bowling

    So, we subtract 5 + 2 + 1 = 8 from 25, leaving us with 17 people who practice only bowling. Write "17" in the remaining part of circle B.

  3. Skiing: 27 people practice skiing in total. We've accounted for:

    • 5 who do all three sports
    • 6 who do running and skiing
    • 1 who does skiing and bowling

    So, we subtract 5 + 6 + 1 = 12 from 27, leaving us with 15 people who practice only skiing. Write "15" in the remaining part of circle C.

The Completed Venn Diagram

Your Venn diagram should now look something like this:

  • Circle A (Running): 21 (only running) + 2 (running and bowling) + 6 (running and skiing) + 5 (all three) = 34
  • Circle B (Bowling): 17 (only bowling) + 2 (running and bowling) + 1 (skiing and bowling) + 5 (all three) = 25
  • Circle C (Skiing): 15 (only skiing) + 6 (running and skiing) + 1 (skiing and bowling) + 5 (all three) = 27

We've successfully constructed and named our Venn diagram! Now, let's move on to the second part of the problem.

Determining Those Who Practice No Activities

Calculating Total Participants

To find out how many people do not participate in any of the three activities, we first need to calculate the total number of people who do participate in at least one activity. We can do this by adding up all the numbers inside the circles of our Venn diagram:

21 (only running) + 17 (only bowling) + 15 (only skiing) + 2 (running and bowling) + 6 (running and skiing) + 1 (skiing and bowling) + 5 (all three) = 67

So, 67 people participate in at least one of the three activities.

Finding Non-Participants

We know there are 70 people in total. To find the number of people who do not participate in any of the activities, we subtract the number of participants from the total number of people:

70 (total people) - 67 (participants) = 3

Therefore, 3 people do not practice any of the three activities.

Final Answer

And there we have it! We've successfully solved the problem.

A. We constructed a Venn diagram representing the number of people participating in running, bowling, and skiing, including those participating in multiple activities.

B. We determined that 3 people do not participate in any of the three activities.

This problem highlights the power of Venn diagrams in visualizing and solving set theory problems. By breaking down the information and filling in the diagram step-by-step, we can easily find the answers we need. Great job, guys! You've tackled this Venn diagram problem like pros!

If you found this explanation helpful, give it a thumbs up and share it with your friends. And if you have any other questions or topics you'd like us to cover, drop them in the comments below. Keep practicing, and you'll become a math whiz in no time!