Solving a Venn Diagram Problem with 50 Students Considering Both Math and Biology

In a classroom of 50 students, 27 students like Math, and 32 students like Biology. Every student has at least one subject they enjoy. To solve this intriguing problem, we can use a Venn diagram with set theory to determine how many students like each subject individually and those who like both subjects. Let's dive into the step-by-step process.

Defining the Sets and Using the Total Students

First, we define our sets:

M represents the set of students who like Math, with M 27. B represents the set of students who like Biology, with B 32. The total number of students is 50, and since every student likes at least one subject, we have:

M ∪ B 50.

Applying the Principle of Inclusion-Exclusion

The principle of inclusion-exclusion states that:

M ∪ B M B - M ∩ B.

Now, let's plug in the values we know:

50 27 32 - M ∩ B.

To solve for the intersection, we follow these steps:

50 59 - M ∩ B.

Rearranging the equation, we get:

M ∩ B 59 - 50 9.

This means that 9 students like both Math and Biology.

Finding Students Who Like Only One Subject

To find the number of students who like only Math:

Students who like only Math M - M ∩ B 27 - 9 18.

And to find the number of students who like only Biology:

Students who like only Biology B - M ∩ B 32 - 9 23.

Summary of the Result

Here is a summary of the results:

Students who like only Math: 18 Students who like only Biology: 23 Students who like both subjects: 9

Venn Diagram Representation

A Venn diagram can visually represent this distribution:

Draw two overlapping circles: one for Math (M) and one for Biology (B). Place the number 9 in the overlapping section (intersection) of both circles. In the Math-only section, place the number 18 because 27 - 9 18. In the Biology-only section, place the number 23 because 32 - 9 23.

The Venn diagram helps you clearly see the distribution of students who like Math and Biology.

Unless this is a trick question and not from a Math course, the solution should be fairly straightforward. This method effectively solves the problem, allowing you to understand the individual and combined preferences of students in the classroom.