Solving the Venn Diagram Puzzle: Understanding Students' Preferences for Math and Science

In this article, we explore a classic problem involving the preferences of 110 students for mathematics and science through the lens of Venn diagrams. We will provide a detailed breakdown of the problem's solution, emphasizing the use of mathematical equations and logical reasoning.

Problem Statement and Analysis

The problem presented is a classic example of set theory and Venn diagrams. It involves 110 students, with some liking mathematics (M), some liking science (S), and some liking both. Given that 25 students like both mathematics and science (M ∩ S), and 40 students like both science and mathematics (S ∩ M), we aim to find the number of students who like mathematics (M).

Mathematical Formulation

Let's define our variables as follows:

x number of students who like mathematics (M) y number of students who like science (S)

The equations provided in the problem can be summarized as:

0.25x y (25% of the mathematics students also like science) 0.40y x (40% of the science students also like mathematics)

Step-by-Step Solution

Starting with the equations:

0.25x y 0.40y x

Substituting y from equation (1) into equation (2):

0.40(0.25x) x

0.1 x

0.10 1

From this, we get:

x 10

This means that 10 students like mathematics.

Alternative Approach with Venn Diagrams

Another approach involves using the ratios derived from the given percentages.

X  Students who like math only
Y  Students who like science only
Z  Students who like both
110  X   Y   Z
Z  0.25X   0.40Y
110  1.25X   1.4Y
X  110 - 1.4Y / 1.25
X  60  Y  25  Z  25

This solution yields whole numbers and realistically divides the students into groups based on their preferences.

Conclusion

The solution to this problem is that 80 students like mathematics (M). This conclusion is reached by understanding the relationships between the students who like both subjects and those who like only one subject. By using algebraic equations and logical reasoning, we can effectively solve the puzzle of the students' preferences in mathematics and science.

In summary, the problem involves understanding the relationships between different groups of students and their overlapping preferences using mathematical equations and Venn diagrams. The solution provides a clear and detailed explanation of how to divide the students based on their preferences.

Additional Insights

The problem illustrates the importance of Venn diagrams in understanding overlapping sets and ratios. It also highlights how set theory and algebraic equations can be used to solve real-world problems related to preferences and groupings.