# Problem Solving With Venn Diagrams

Call one circle people who have gone to Punta de las Cuevas and the other circle people who have gone to Salt Bay.

Number of people who use Television :n(T) = 115Number of people who use Radio : n(R) = 110Number of people who use Magazine : n(M) = 130Number of people who use Television and Magazinesn (Tn M) = 85Number of people who use Television and Radio :n(Tn R) = 75Number of people who use Radio and Magazine :n(Rn M) = 95Number of people who use all the three : n(Tn Rn M) = 70 At a certain conference of 100 people there are 29 Indian women and 23 Indian men.

Out of these Indian people 4 are doctors and 24 are either men or doctors. Find the number of women doctors attending the conference.

The Salt Bay circle should have 3 people in it, but there are already two people in the part of the circle that intersects with the other circle, so there is just one person in the part of the Salt Bay circle that is not within the other circle.

Similarly, there will be 2 people in the Punta de las Cuevas circle that are not in the part that overlaps the Salt Bay circle. This is typical of many more complicated problems, in that the key idea is subtraction.

When he asked how many had been to Punta de las Cuevas, four children raised their hands.

When he asked how many had been to Salt Bay, three raised their hands.We know the whole of a set, and one part, so we subtract to find the other part.Simultaneously, Doctor Keith answered, without using a Venn diagram, showing that this is just a tool for organizing information, not an essential part of the work: This is an example of everyone's favorite type of problem, the word problem.Let’s look at one more example with two sets, this one using a table rather than a Venn diagram.This is from 2009: Alternate Solution Path to Venn Diagram Problem There are 30 students in a math class. I know that the correct answer (according to the book) is 13. I have either 10 or 7 as my answers, not knowing if the "both club" members are included already in the listed number of members.The two students who have been to both parks would have raised their hands for both Punta de las Cuevas and Salt Bay.Thus two of the students in the Punta de las Cuevas group are also in the Salt Bay group. For Punta de las Cuevas, four students have been but two also went to Salt Bay, so we have to remove them (remember we want students who have only been to Punta de las Cuevas). of students who have only been to Punta de las Cuevas Thus only two students have only been to Punta de las Cuevas.In a town 85% of the people speak Tamil, 40% speak English and 20% speak Hindi.Also 32% speak Tamil and English, 13% speak Tamil and Hindi and 10% speak English and Hindi, find the percentage of people who can speak all the three languages.Similarly, for Salt Bay we have three students who have gone, but two also went to Punta de las Cuevas so we must remove those two from the three who have been. of students who have only been to Salt Bay Thus only one student has gone to only Salt Bay.To get the final answer we note that the sum of the students who have only been to Punta de las Cuevas, plus those who have only been to Salt Bay, plus those who have been to both must be the total in the group.