In an examination, 80 students secured first class marks in English or Mathematics. Out of these, 50 students obtained first class marks in Mathematics and 10 students in English and Mathematics both. How many students secured first class marks in English only?

To determine how many students secured first class marks in English only, we can break down the problem step-by-step using set theory concepts.
Let's define:

Let E represent the set of students who secured first class marks in English.
Let M represent the set of students who secured first class marks in Mathematics.

Given:

The total number of students who secured first class in either English or Mathematics is 80. This can be represented as the union of the two sets, ∣ E ∪ M ∣ = 80 .
The number of students who secured first class in Mathematics is 50, which means ∣ M ∣ = 50 .
The number of students who secured first class in both subjects, English and Mathematics, is 10, so ∣ E ∩ M ∣ = 10 .

We are asked to find the number of students who secured first class marks in English only. This is represented by ∣ E ∣ − ∣ E ∩ M ∣ .
To find ∣ E ∣ , we use the principle of Inclusion-Exclusion:
∣ E ∪ M ∣ = ∣ E ∣ + ∣ M ∣ − ∣ E ∩ M ∣
Substituting the given values:
80 = ∣ E ∣ + 50 − 10
Simplifying the equation:
80 = ∣ E ∣ + 40
∣ E ∣ = 80 − 40 = 40
Therefore, the number of students who secured first class marks in English only is:
∣ E ∣ − ∣ E ∩ M ∣ = 40 − 10 = 30
So, 30 students secured first class marks in English only.

Answered by DanielJosephParker | 2025-07-22