Calculating Success Rates Among Students in English and Mathematics

Understanding the distribution of student success and failure rates in specific subjects is crucial for educational institutions. This article explores how to calculate the number of students who passed in both English and Mathematics, using a principle known as the principle of inclusion-exclusion. Let's delve into the details.

Given Data and Initial Calculations

Consider the scenario where out of 100 students, 50 failed in English, 30 failed in Mathematics, and 12 failed in both subjects. We aim to determine the number of students who passed in both subjects.

We can represent the failure rates as probabilities of failing in at least one subject. Let's denote:

n - E number of students who failed in English 50 n - M number of students who failed in Mathematics 30 n - B number of students who failed in both subjects 12

Using the principle of inclusion-exclusion, the number of students who failed in at least one subject is:

[n E M - B 50 30 - 12] [n 68]

Thus, 68 students failed in at least one of the subjects. Consequently, the number of students who passed in both subjects is:

[100 - 68 32]

Therefore, 32 students passed in both English and Mathematics.

Another Example with Different Numbers

Let's consider another scenario where out of 300 students:

70 failed in Mathematics 80 failed in English

Here, the number of students who failed in at least one subject can be calculated as:

[text{Number of students who failed in at least one subject} 70 80 - (text{students who failed in both})]

If 30 students passed in Mathematics, then the number of students who failed in Mathematics or English or both is:

[150 - 30 120]

Hence, the number of students who passed in both subjects is:

[300 - 120 180]

Therefore, 180 students passed in both Mathematics and English.

Further Scenario with Additional Data

Suppose we have the following additional data:

50 students failed in Account (let's denote this number as X/2) 30 students failed in English (let's denote this number as 3X/10) 25 students failed in both subjects (let's denote this number as X/4)

We can set up the equation based on the principle of inclusion-exclusion:

[frac{X}{4} 225]

Solving for X:

[X 900]

So, the total number of students is 900. The number of students who failed in Account is:

[900 / 2 450]

The number of students who only failed in Account is:

[450 - 225 225]

The number of students who only failed in English is:

[450 - 270 180]

Hence, the number of students who passed in both subjects is:

[900 - 630 270]

Conclusion

By applying the principle of inclusion-exclusion and carefully analyzing the given data, we can accurately determine the success rates of students in both English and Mathematics. This approach is essential for educational institutions to understand student performance and allocate resources effectively.