Understanding the Intersection of Math and English Performance Among Students

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In academic evaluations, it is common to encounter questions that require understanding and applying set theory to real-world scenarios. One such typical question involves determining the number of students who failed only English given certain conditions. Let's break down the problem step by step to find a solution.

Context and Given Information

In this problem, we are given the following data:

Total number of students 60 Number of students who passed English 30 Number of students who passed Math 42

The task is to find the number of students who failed only English. However, solving this problem requires a clear understanding of the relationships between the sets of students who passed or failed each subject.

Analysis and Solution

Firstly, it is important to note that the success or failure in one subject is not necessarily independent of the success or failure in the other. Therefore, we need to use the principle of inclusion-exclusion to solve this problem.

Let's define the following sets:

A: Set of students who passed English B: Set of students who passed Math

We have the following information:

|A| (number of students who passed English) 30 |B| (number of students who passed Math) 42 Total number of students, |U| 60

Z: The number of students who passed both subjects is denoted as |A ∩ B|.

According to the principle of inclusion-exclusion:

|A ∪ B| |A| |B| - |A ∩ B|

We know that the total number of students is 60, and all students have taken at least one of the two subjects. Therefore:

|A ∪ B| 60

Substituting the known values:

60 30 42 - |A ∩ B|

|A ∩ B| 12

Now, we need to find the number of students who failed only English. This can be determined by subtracting the number of students who passed both subjects from the number of students who passed English:

Number of students who passed English only |A| - |A ∩ B| 30 - 12 18

Therefore, the number of students who failed English but passed Math is:

Total students - (Students who passed both subjects Students who passed English only)

60 - (12 18) 60 - 30 30 - 18 12

Thus, 12 students failed only English.

Conclusion

Through a step-by-step analysis, we have determined that 12 students failed only English. This problem highlights the importance of understanding the relationships between different sets and the application of inclusive-exclusion principles in solving such problems.

Related Keywords

student performance math and English homework assistance

For a deeper understanding of similar problems, refer to resources on set theory and problem-solving in academic assessments.