Venn Diagram Word Problems


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Venn Diagram Word Problems

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  2. 2. Venn Diagram Problems can be challenging for beginners.It is extremely important to:Read the question carefully and note down all keyinformation, usually summarised in dot point form.Know the standard parts of a Venn DiagramWork in a step by step mannerCheck at the end that all the numbers add up properly.
  3. 3. "A class of 28 students were surveyed and asked if theyever had cats or dogs for pets at home. • 8 students said they had only ever had a dog. • 6 students said they had only ever had a cat. • 10 students said they had a cat and a dog. • 4 students said they had never had a cat or a dog.“ Since this question is about cats and dogs, it will require a two circle Venn Diagram. (See Next Slide)
  4. 4. Our problem is an easy one, where we have been given allof the numbers for the items required on the diagram.We do not need to work out any missing values.All we need to do is place the numbers from the wordproblem onto the standard Venn Diagram and we are done.Eg. Cats Only = 6, Dogs Only = 8, Both Cats & Dogs = 10, and Neither Cat or Dog = 4 (See Next Slide for completed diagram)
  5. 5. The answer for this question will actually be the same asthe Cats and Dogs in Question 1. However this time we aregiven less information, and have to do some working out."A class of 28 students were surveyed and asked if theyever had dogs or cats for pets at home. • 18 students said they had a dog. • 16 students said they had a cat. • 4 students said they had never had a dog or a cat."
  6. 6. Word Problem Two does not contain the word "only"anywhere in it, and this is an indication that we will needto do some working out.The question states that: "18 students said they had a dog“ (without the word "only" anywhere in there).This means that the total of the Dogs circle is 18.The total 18 students for Dogs includes people thathave both a cat and a dog, as well as people who onlyhave a dog.Some people will not read this question carefully, and willincorrectly take the supplied numbers and put them straightinto a Venn Diagram like this. (See Next Slide)
  7. 7. Always check at the end that the numbers add up to the "E"Grand Total. Eg. 16 + 18 + 4 = 38 is much bigger than the"E" everything total which is 28 students.This “E” Total is smaller because some students have botha cat and a dog. We have not accounted for this yet.Other people might think that we do not have enoughinformation given to us, and therefore it is impossible to dothis problem.Lets put down on our diagram all of the information that weknow. (See Next Slide)
  8. 8. All we have left to work out is the number of Cats and Dogsfor the “Intersection” at the centre of the diagram.We can do this any one of three ways:Cats and Dogs = Total Cats - Only CatsORCats and Dogs = Total Dogs - Only DogsORCats and Dogs = E Total - Only Cats - Only Dogs - (No catsand No Dogs)Any way that we work it out, the answer is 10.So we now have our Diagram completed. (See Next Slide)
  9. 9. Work out What Information is given, and what needs to becalculated.Circles Total = E everything - (No Cats and No Dogs)Cats Only = Circles Total - Total DogsDogs Only = Circles Total - Total CatsCats and Dogs = Cats Total - Cats OnlyFinally, check that all the numbers in the diagram add up toequal the "E" everything total.
  10. 10. "Fifty people were surveyed and only 20 people said that theyregularly eat Healthy Foods like Fruit and Vegetables.Of these 20 healthy eaters, 12 said they ate Vegetables every day."Draw a Venn Diagram to represent these results."This problem is quite different to our other two circle diagrams.Cats and Dogs are quite different, and needed two separate circles.However Healthy Foods and Vegetables are not different to each otherbecause Vegetables are a type of Healthy Food.We say that vegetables are a "Subset" of Healthy Foods.This means that we do not separate the circles, and we draw our circlesinside each other like this. (See Next Slide)
  11. 11. "Draw a Venn Diagram which divides the twelve months of the yearinto the following two groups: • Months whose name begins with the letter "J“, • Months whose name ends in "ber" .You will need a two circle Venn Diagram for your answer.”Months starting with J = { January, June, July }Months ending in "ber" = { September, October, November, December }The two sets do not have any items in common, and so we will notneed to overlap them. They are “Mutually Exclusive” or “Disjoint”.The remaining months will need to go outside of our two circles.There should be all twelve months in the diagram when we are finished.
  12. 12. The working out steps for harder problems are:Work out What Information is given, and what needs to be calculated.Check to see if the two sets are "Subsets" or "Disjoint" sets.If they are "Intersecting Sets" then some of the followingcalculations may be required:Circles Total = E everything - (Not in A and Not in B)In A Only = Both Circles Total - Total in BIn A Only = The A Circle Total - Total in the intersection (A and B)In B Only = Both Circles Total - Total in AIn B Only = The B Circle Total - Total in the intersection (A and B)In the Intersection (A and B) = Total in B - In B OnlyIn the Intersection (A and B) = Total in A - In A Only Finally, always check that the numbers in the diagram all add up to equal the "E" everything total.
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