How to Solve Venn Diagram Problems Part 2

In the previous posts, the introduction and the second part of this series, we have learned the basics of Venn Diagrams as well as solving the 2-circle Venn Diagram problem. In this post, we are going to solve a more complicated problem which is composed of 3-circle Venn diagram problem.

Venn Diagram Problem

There are 100 students surveyed and asked which of the following subjects they take this semester: Mathematics, English, or Biology. Below is the result of the survey.

  • 35 responded English
  • 50 responded Mathematics
  • 29 responded Biology
  • 12 responded Mathematics and English
  • 8 responded English and Biology
  • 11 responded Biology and Math
  • 5 responded all


  1. How many students are not taking any of the three subjects?
  2. How many students take Math, but not Biology or English?
  3. How many students take Math and English, but not Biology?


As we have done in Solving Venn Diagram Problems Part 1, we started with the overlapping circles. In this problem, we have four regions which overlap. The the easiest strategy is to start at the center, the part where the three circles overlap. In short, we start from bottom to the top in the result above.

(1). Five students responded that they took all the subjects, so we put 5 at the center.

Venn Diagram Problems

(2). Eleven responded Biology and Math. So, we should put it in the Biology-Math overlap. However, of the 11 who takes Biology and Math, 5 were also taking English as shown in (1). So, there are 11-5 = 6 students in the Math-Biology overlap.

Venn Diagram Problems

(3). Eight responded Biology and English. But of those 8 taking Biology and English, 5 are also taking Math. So, there are 8 – 5 = 3 students who are taking Biology and English.

Also, there are 12 students who are taking Math and English, and of those 12, 5 are also taking Biology, so there are 12 – 5 = 7 who are taking Math and English. Now see the next figure to see how the Venn diagram should look like after this step.

Venn Diagram Problems


(4). Next, 29 students responded Biology. But notice that 6, 5, and 3 are already in the Biology circle. So, we subtract those students from 29. That is, 29 – (6 + 5 + 3) = 29 – 14 = 15. So, there are 15 students who take only Biology.

Venn Diagram Problems

(5). Lastly, there are 50 students who are taking Math and 35 who are taking English. But in the Math circle there are 6 + 5 + 7 students who are also taking the other subjects and in the English circle, there are 3 + 5 + 7 students who are also taking the other subjects. Therefore, we can have the following calculations

Number of Students Who Take Only Math = 50 – (6 + 5 + 7) = 32

Number of Students Who Take Only English = 35 – (3 + 5 + 7) = 20

After answering the problems above, the Venn diagram should look like below.

Venn Diagram ProblemsNow, to answer the first question, how many students did not take any of the three subjects, recall that there are 100 students who were surveyed. If we add all the numbers in the diagram, 15 + 32 + 20 + 6 + 7 + 3  + 5, the sum is only 88. Therefore, 100 – 88 = 12 students did not take any of the three subjects.

Venn Diagram Problems

For question 2, another way to rephrase it is how many students take only Math, so the answer is 32.

For question 3, the answer is 7 (can you see why?)

Now, that’s it for the Venn Diagram series. We will have practice problems in the next posts.

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