Solve problems using Venn diagrams
You may be asked to solve problems using Venn diagrams in your exam. It is really important you draw the Venn diagram and add information as you go along. This will help you keep an overview of what is going on.
Example
- 100 visitors to Festival completed a questionnaire
- All 100 visitors had seen at least one of the following bands – Band X, Band Y and Band Z
- 14 of the visitors had seen Band X and Band Y and, of these, 3 had seen all bands
- 36 people had seen Band X
- 55 people had seen Band Y
- 53 people had seen Band Z
- Some further information is given on the Venn diagram below
How many visitors had seen Band X but not Band Y or Band Z?
Solution
First of all, work out how many people have seen Band X and Band Y only:
14 – 3 = 11
11 visitors have seen Band X and Band Y only. We can now fill this in on our diagram.
If we now look at Band Y circle, we have all the sections bar one.
Now add up the numbers we have so far in Band Y circle:
20 + 11 + 3 = 34.
You can see we have 34 people so far, but we know 55 people saw Band Y:
55 – 34 = 21.
21 people are remaining and go in the blank section. Therefore, 21 people have seen Band Y and Band Z.
If we now look at Band Z circle, we have all the sections bar one. If we now add up the numbers we have so far in Band Z's circle:
23 + 21 + 3 = 47
You can see we have 47 people so far, but we know 53 people saw Band Z:
53 – 47 = 6
So 6 people are left over to go in the section that is blank. 6 people have seen Band Z and Band X, we can now fill this in on our Venn diagram.
If we now look at Band X circle, we have all the sections bar one. Let’s add up the numbers we have so far in this circle:
11 + 6 + 3 = 20
You can see we have 20 people so far, but we know 36 people saw Band X:
36 – 20 =16
So 16 people are left over to go in the section that is blank. We can now fill this in on our Venn diagram.
16 visitors had seen Band X, but not Band Y or Band Z.
