Three Circle Venn Diagrams

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Three Circle Venn Diagrams

  1. 1. Image purchased by Passy’s World from istockphoto
  2. 2. Three Circle Venn Diagrams can take quite a bit of working out.The steps to follow are generally these:Get the Information from the Question that can go straight onto theVenn Diagram and place it there.Work through the remaining information, a bit at a time, to work outeach of the missing values.Work from the centre of the diagram outwards when finding theseunknown values.Check that all the numbers on the final diagram add up to theE = everything grand total. We will start by reading a completed three circle diagram.
  3. 3. Dance 20 14 5 Rock 8 2 16 5RAP Free ClipArt Images were obtained from Google Images
  4. 4. Question 1 – Total People Dance 20 What is the Total Number of People represented in this Diagram ? 14 5 ANSWER: Add up all the numbers on 8 the diagram and the total is 70 . 2 16 5RAP
  5. 5. Question 2– Rock People Danc What is the Total Number of People 20 e who like Rock music ? ANSWER: Add up 14 5 Rock all the numbers in the Rock Circle, 8 and the total is 31 . 2 16 5 RAP
  6. 6. Question 3 – Probability Dance 20 If one person is chosen at random, what are the odds of picking a person who likes Rock ? 14 5 ANSWER: 31 out 8 of 70 people liked Rock, so odds are: 2 16 Pr(Rock) = 31 / 70 5 or 31 out of 70.RAP Rock
  7. 7. Question 4 – All Types How many of the People like all 20 Dance three types of music : Dance, Rock, and Rap ? 14 5 ANSWER: 8 people as shown in the 8 diagram’s centre, where all three 2 16 circles overlap. 5 RAP Rock
  8. 8. Question 5 – Probability If a person is chosen randomly, Dance what are the odds that they like all three types of 20 music ? ANSWER: 8 people out of the total 70 14 5 liked all three, so odds are 8 / 70. 8 2 16 5RAP Rock
  9. 9. A Class of 40 students completed a survey on what pets they like.The choices were: Cats, Dogs, and Birds.Everyone liked at least one pet.10 students liked Cats and Birds but not dogs6 students liked Cats and Dogs but not birds2 students liked Dogs and Birds but not Cats2 students liked all three pets10 students liked Cats only9 students liked Dogs only1 student liked Birds onlyRepresent these results using a three circle Venn Diagram.The type of Venn Diagram we need to use is shown on the next slide.
  10. 10. = everything Birds Cats Cats Only & Birds Birds Only Not Dogs Cats Birds DogsCats Cats & Dogs Not Birds Birds & Dogs Not Cats Dogs Only Dogs
  11. 11. This three circle word problem is an easy one.All of the number values for each section of the diagramhave been given to us in the question.All we need to do is carefully put the number values ontothe Diagram.We also need to check that all of the numbers add up tothe total of 40 students when we are finished.The completed Venn Diagram is shown on the next slide.
  12. 12. = 40 students Birds 10 10 1 2Cats 6 2 9 Dogs
  13. 13. = 40 students Birds 10 10 1 2Cats 6 2 9 Dogs
  14. 14. This is a harder version of Problem One, where we are given lessinformation in the question text. This means that we will need to dosome working out steps to get to the final completed diagram.“A Class of 40 students completed a survey on what pets they like.The choices were: Cats, Dogs, and Birds.Everyone liked at least one pet.10 students liked Cats and Birds but not Dogs2 students liked Dogs and Birds but not Cats12 students liked Cats and Birds8 students liked Cats and DogsAll together, 28 students liked Cats, 19 students liked Dogs, and15 students liked Birds.Represent these results using a three circle Venn Diagram.” The type of Venn Diagram we need is shown on the next slide.
  15. 15. = everything Birds Cats Cats Only & Birds Birds Only Not Dogs Cats Birds DogsCats Cats & Dogs Not Birds Birds & Dogs Not Cats Dogs Only Dogs
  16. 16. Let’s look carefully at the information we have been given“A Class of 40 students” - This is the E = everything total and can go on diagram now“Everyone liked at least one pet” – There is nobody outside the circles“10 students liked Cats and Birds but not Dogs” – This can go on the diagram now“12 students liked Cats and Birds” – This can help us work out values later“ 8 students liked Cats and Dogs” – This can help us work out values later2 students liked Dogs and Birds but not Cats - This can go on the diagram nowAll together, 28 students liked Cats, 19 students liked Dogs, and 15 liked Birds.- These three values are the Totals of each of the circles on our diagram. We will use these later to help work out the number values for: “Cats Only”, “Dogs Only” and “Birds Only” The next slide shows our Venn Diagram so far.
  17. 17. = 40 students Birds Cats Only 10 Birds Only Total Cats 15 Birds DogsCats Cats & Dogs Not Birds 2Total 28 Dogs Only Dogs Total 19
  18. 18. Let’s now work on this piece of information:“ 12 students liked Cats and Birds”This tells us that : the Cats and Birds Not Dogs part of the diagram + the Cats and Birds and Dogs part in the centre = A total of 12 items.( We only care that the students liked Cats and Birds, and we do not care whether or not they liked Dogs. ) The next slide highlights these areas on our Venn Diagram .
  19. 19. = 40 students Birds Cats Only 10 LIKES CATS AND BIRDS Cats Question says - In total there are 12 people Birds who like Cats and Birds. DogsCats 2 This means that the number of people who like “Cats and Birds and Dogs” in the very centre of our diagram works out as the value of “2”. Dogs Only Dogs
  20. 20. = 40 students Birds Cats Only 10 Birds Only 2Cats Cats & Dogs Not Birds 2 Dogs Only Dogs
  21. 21. = 40 students Birds WORKING OUT BIRDS ONLY Cats Only 10 Birds Only Total We know the Total of the Birds Circle is 15. = ? 15 This means that 2Cats 10 + 2 + 2 + ? = 15 2 For this to be true “Birds Only” needs to equal 1. Dogs Only Dogs
  22. 22. = 40 students Birds Cats Only 10 Total 1 15 2Cats Cats & Dogs Not Birds 2 Dogs Only Dogs
  23. 23. Let’s now work on this piece of information:“ 8 students liked Cats and Dogs”This tells us that : the Cats and Dogs Not Birds part of the diagram + the Cats and Dogs and Birds part in the centre = A total of 8 items.( We only care that the students liked Cats and Dogs, and we do not care whether or not they liked Birds. ) The next slide highlights these areas on our Venn Diagram .
  24. 24. = 40 students Birds Cats Only 10 1 LIKES CATS AND DOGS Question says - In total 2 there are 8 people who like Cats and Dogs.Cats Cats & Dogs Not Birds 2 This means that People who like “Cats and Dogs and Not Birds” needs to equal 6. 2 + 6 = 8 Total for the Dogs Only yellow Cats and Dogs area. Dogs
  25. 25. = 40 students Birds Cats Only 10 1 2Cats 6 2 Dogs Only Dogs
  26. 26. = 40 students Birds Cats Only =? 10 WORKING OUT CATS ONLY 1 We know the Total of the Cats Circle is 28. 2 This means thatCats 6 2 10 + 2 + 6 + ? = 28 For this to be trueTotal “Cats Only” needs to equal 10. 28 Dogs Only Dogs
  27. 27. = 40 students Birds 10 10 1 2Cats 6 2 Dogs Only Dogs
  28. 28. = 40 students BirdsWORKING OUT DOGS ONLY 10 10 1We know the Total ofthe Dogs Circle is 19.This means that 2 Cats6 + 2 + 2 + ? = 19 6 2For this to be true“Dogs Only” needs to equal 9. Dogs Only =? Dogs Total 19
  29. 29. = 40 students Birds 10 10 1 2Cats 6 2 9 Dogs
  30. 30. = 40 students Birds 10 10 1 2Cats 6 2 9 Dogs
  31. 31. Three Circle Venn Diagrams can take quite a bit of working out.The steps to follow are generally these:Get the Information from the Question that can go straight onto theVenn Diagram and place it there.Work through the remaining information, a bit at a time, to work outeach of the missing values.Often we work from the centre of the diagram outwards, when findingthese unknown values.Check that all the numbers on the final diagram add up to theE = everything grand total.
  32. 32. http://passyworldofmathematics.com/All slides are exclusive Copyright of Passy’s World of Mathematics

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