Question

1. in a music school of 124 students, 82 students play the piano, 71 students play the guitar and 10 students play neither.

a =

b =

c =

d =

Explanation:

Step1: Find number of students who play at least one instrument

Total students = 124, students who play neither = 10. So students who play at least one instrument: \(124 - 10 = 114\)

Step2: Find number of students who play both instruments (B)

Let \(P\) be piano players (\(n(P)=82\)), \(G\) be guitar players (\(n(G)=71\)). Using \(n(P \cup G)=n(P)+n(G)-n(P \cap G)\), we have \(114 = 82 + 71 - n(P \cap G)\). Solving: \(n(P \cap G)=82 + 71 - 114 = 39\). So \(B = 39\)

Step3: Find number of students who play only piano (C)

\(C=n(P)-n(P \cap G)=82 - 39 = 43\)

Step4: Find number of students who play only guitar (D)

\(D=n(G)-n(P \cap G)=71 - 39 = 32\)

Step5: A is students who play neither, so \(A = 10\)

Answer:

\(A = 10\)
\(B = 39\)
\(C = 43\)
\(D = 32\)