Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all three subjects, while twenty study H and E, but not P. Every student who studies P also studies H or E or both. If the number of students studying H equals that studying E, then the number of students studying H is
Correct Answer: 52
Solution and Explanation
Step 1: Define variables
Let:
- \( h \) = students studying only H
- \( e \) = students studying only E
- \( x \) = students studying only H and P but not E
- \( y \) = students studying only E and P but not H
Given:
- Only P = 0
- All three subjects = 10
- Only H and E but not P = 20
Step 2: Equating total H and E counts
Number of students studying H: \[ H = h + x + 20 + 10 \] Number of students studying E: \[ E = e + y + 20 + 10 \] Given \( H = E \), we have: \[ h + x = e + y \]
Step 3: Total students
Total students: \[ h + x + 20 + 10 + e + y = 74 \] Substitute \( h + x = e + y \): \[ (h + x) + (e + y) + 30 = 74 \] \[ (h + x) + (h + x) = 44 \] \[ 2(h + x) = 44 \] \[ h + x = 22 \]
Step 4: Number studying H
\[ H = (h + x) + 20 + 10 = 22 + 20 + 10 = 52 \]
✅ Final Answer: The number of students studying H = 52
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