Question

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

Accepted Answer

52. Total = 74. Let regions be: $n(H \cap E \cap P) = 10$. $n(H \cap E \cap P') = 20$. Since every student studying P also studies H or E or both, $n(	ext{only } P) = 0$. Let $n(	ext{only } H) = h$, $n(	ext{only } E) = e$. Let $n(H \cap P \cap E') = a$, $n(E \cap P \cap H') = b$. Sum = $h + e + 20 + 10 + a + b = 74 \Rightarrow h + e + a + b = 44$. $n(H) = h + 20 + 10 + a = h + a + 30$. $n(E) = e + 20 + 10 + b = e + b + 30$. Since $n(H) = n(E) \Rightarrow h + a = e + b$. Thus $h + a = 22$. $n(H) = (h + a) + 30 = 22 + 30 = 52$.

Three Subject Venn Diagram

Explanation

Practise this under exam conditions