Question

There are 80 students admitted to a Music School. Some students can play only violin and some can play only harmonium. 20 children can play both violin and harmonium. If the number of students who can play violin is 30, then how many students can play harmonium? Also, how many can play only harmonium and how many can play only violin?

• A. 70, 50 and 10 respectively
• B. 60, 45 and 12 respectively
• C. 65, 50 and 15 respectively
• D. 68, 48 and 10 respectively
• E. 72, 45 and 11 respectively

Hint:

Venn diagrams are the most efficient way to visualize overlapping sets in music or language problems.

Answer 1

The Correct Option is A

Concept: Set Theory: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
Step 1:Identify the values.
Total students $n(V \cup H) = 80$. Students who play both $n(V \cap H) = 20$. Students who play violin $n(V) = 30$.

Step 2: Calculate only Violin and only Harmonium.

• Only Violin = $n(V) - n(V \cap H) = 30 - 20 = 10$.
• Only Harmonium = Total - $n(V) = 80 - 30 = 50$.
• Total Harmonium $n(H)$ = Only Harmonium + Both = $50 + 20 = 70$. The values are: Harmonium (70), Only Harmonium (50), Only Violin (10).