Solving the Overlapping Subject Problem: A Venn Diagram Approach

When faced with a problem involving the overlap between two subjects in a class setting, using a Venn diagram can be a straightforward and effective method. This article explains how to approach such a problem through the lens of set theory. We will walk through the steps to solve a specific math problem, demonstrate the calculations, and leverage the principle of Venn diagrams.

Problem Statement

There are 30 students in a class. 15 study woodwork, and 13 study metalwork. 6 students study neither of the two subjects. How many students study woodwork and not metalwork?

Using a Venn Diagram

A Venn diagram is a pictorial tool used to represent set relationships. In this case, we have two sets of students: those who study woodwork and those who study metalwork. The diagram would look something like this, with the region outside the circles representing students who study neither subject:

In the diagram, the regions are as follows:

W: Students who study only woodwork. M: Students who study only metalwork. B: Students who study both woodwork and metalwork.

The problem statement gives us the following information:

Number of students in the class: 30 Number of students who study neither woodwork nor metalwork: 6 Number of students who study woodwork: 15 Number of students who study metalwork: 13 Number of students who study both subjects: B

Using this information, we can set up the following equations:

Total number of students Number of students who study woodwork Number of students who study metalwork - Number of students who study both subjects Number of students who study neither subject

30 15 13 - B 6

Solving for B:

30 34 - B

B 4

So, 4 students study both woodwork and metalwork.

Calculating Students Studying Woodwork Only

To find the number of students who study woodwork only (i.e., W), we subtract the number of students who study both subjects (B) from the number of students who study woodwork:

W 15 - 4 11

Alternative Solutions

The same problem can be approached using different methods, all of which lead to the same conclusion:

Number of students who study at least one subject Total students - Number of students who study neither subject 30 - 6 24

Number of students who study both subjects (B) (Number of students who study woodwork Number of students who study metalwork) - Number of students who study at least one subject (15 13) - 24 4

Number of students who study woodwork only Number of students who study woodwork - Number of students who study both subjects 15 - 4 11

Let x be the number of students who study both. Number of students who study only woodwork 15 - x, and number of students who study only metalwork 13 - x. Subtracting the number of students who study neither (6) from the total number of students gives us: (15 - x) (13 - x) x 6 30. Solving for x, we find x 4, thus 11 students study only woodwork.

Let B be the number of students who study both. The total number of students who study woodwork or metalwork or both is 24, so 24 - 13 (metalwork) 11 students who study woodwork only.

Conclusion

Using different problem-solving techniques, we consistently arrive at the same answer: 11 students study woodwork and not metalwork. This approach helps in understanding the intersection and individual components of overlapping sets, which is a fundamental concept in set theory and practical for many real-world scenarios.