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Three circle Venn Diagrams are a step up in complexity from two circle diagrams.

In this lesson we first look at how to read three circle diagrams. We then look at some word problems.

Reading Three Circle Diagrams

A Music Survey was carried out to find out what types of music a group of people liked.

The results were placed into the following three circle Venn Diagram.

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To find out the total number of people surveyed, (or the E = Everything value), we add up all the numbers in the diagram.

When we do this the answer is 70.

This “E = Everything value” is also called “The Universal Set” of everything in the universe of our diagram.

If we want to find the total number of people who like “Rock” music, then we add up all of the numbers in the “Rock” circle, including the areas where “Rock” overlaps with the other circles.

The Total Rock People is: 16 + 2 + 8 + 5 = 31 people.

We can also work out Probability or Odds from our Venn Diagram.

For example we have found that 31 people out of 70 like Rock Music.

So if we pick any one person at random from our group, the chances, or odds, or probability, that they will like Rock music is 31 out of 70, or 31 / 70, or 31/70 x100 = 44%.

We can find the number of people who like all three types of music, by going to the centre of our diagram, where all three circles overlap.

There are 8 people who like all three types of music.

Venn Diagram Word Problem One

This first problem is a fairly easy one, where all of the information we need has been given to us in the question.

“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 dogs

6 students liked Cats and Dogs but not birds

2 students liked Dogs and Birds but not Cats

2 students liked all three pets

10 students liked Cats only

9 students liked Dogs only

1 student liked Birds only

Represent these results using a three circle Venn Diagram.”

The type of three circle Venn Diagram we will need is the following:

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This three circle word problem is an easy one.

All of the number values for each section of the diagram have been given to us in the question.

All we need to do is carefully put the number values onto the Diagram.

We also need to check that all of the numbers add up to the total of 40 students when we are finished.

The completed Venn Diagram is shown below.

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Note that we do not need to color in and fill in the circles on Venn Diagrams.

The following diagram is also correct and a fully acceptable answer.

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Venn Diagram Word Problem Two

This is a harder version of Problem One, where we are given less

information in the question text. This means that we will need to do

some working out steps to get to the final completed diagram.

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We need the exact same type of Venn Diagram as for Question 1.

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We now need to go carefully through the word problem, and work out what information we can use straight away.

There should always be some data we can place onto the diagram to get started, even though it might not seem to be that much.

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When we place what we know so far onto the diagram, this is what we have:

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We now need to work through the other information in the word problem, one piece at a time.

Usually in these problems we need to work on the overlapping parts in the centre of the diagram, and then work our way out to the “Cats Only”, “Dogs Only”, and “Birds Only” outer sections of the diagram.

Remember: Work for the Inside Out.

Here is what we will do next.

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This information refers to the following section of the Venn Diagram.

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We can now fill in the answer of “2” onto the centre of our diagram.

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We now have the Birds circle nearly completed.

The only thing left to do is work out the “Birds Only” section, which we will now do.

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We can now fill in the answer of “1” onto our diagram.

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We now have the Birds circle completed.

Next we work on the Cats Circle, following the exact same steps as we did on the Birds circle.

First we need to work on the overlaps that involve Cats and other animals in the centre of the diagram.

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This information refers to the following section of the Venn Diagram.

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We can now fill in the answer of “6” onto our diagram.

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We are nearly there !

We now only have “Cats Only” and “Dogs Only” to work out, and we have all the information we need to do this.

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We can now place the “Cats Only” answer onto our diagram.

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We now only need to work out “Dogs Only”.

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Our Diagram for Problem Two is now finally complete.

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Note that we do not need to color in and fill in the circles on Venn Diagrams.

The following diagram is also a correct and a fully acceptable answer.

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Venn Diagram Word Problems Summary

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Three Circle Videos

Here is a great Venn Diagrams video which also explains the “Inclusion / Exclusion” method.

This is an interesting three circles problem where they use a table of values to help with the working out.

Here is a video which shows how to do “Complement Sets” for three circle diagrams.

Related Items

Introduction to Venn Diagrams

Venn Diagram Word Problems

Real World Venn Diagrams

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