75 Students secured forst division marks either in English or in Mathematics or in both . If 50 of them secured first division in Mathematics and 10 in both English and Mathematics , then how many got first division in English ?

Answer

To solve the problem step by step, we can use the principle of sets and Venn diagrams.

### Step 1: Define the sets
Let:
- \( E \) = the number of students who secured first division in English
- \( M \) = the number of students who secured first division in Mathematics
- \( B \) = the number of students who secured first division in both English and Mathematics

From the problem, we know:
- Total students (either in English or Mathematics or both) = 75
- Students who secured first division in Mathematics \( M = 50 \)
- Students who secured first division in both subjects \( B = 10 \)

### Step 2: Calculate students who secured first division only in Mathematics
To find the number of students who secured first division only in Mathematics, we subtract those who secured first division in both subjects from the total in Mathematics:
\[ \text{Only in Mathematics} = M - B = 50 - 10 = 40 \]

### Step 3: Set up the equation for total students
The total number of students can be expressed as:
\[ \text{Total} = (\text{Only in English}) + (\text{Only in Mathematics}) + (\text{Both}) \]
Let \( E' \) be the number of students who secured first division only in English. Thus, we can write:
\[ 75 = E' + 40 + 10 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 75 = E' + 50 \]
Subtract 50 from both sides:
\[ E' = 75 - 50 = 25 \]

### Step 5: Calculate total students who secured first division in English
Now, to find the total number of students who secured first division in English, we add those who secured only in English and those who secured in both:
\[ E = E' + B = 25 + 10 = 35 \]

### Final Answer
Thus, the number of students who got first division in English is **35**.

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