Question

1. in mr. rojo’s music class, 14 students play piano, 18 students play guitar, and 9 play drums. no student plays any other instruments. the following information is also true:

7 students play only guitar;
2 students play all three instruments;
5 students play only piano;
1 student plays piano and drums, but not guitar

this information is placed in a venn diagram as shown (piano, drums, guitar venn diagram with 5, 1, 2, 7 labeled).

which of the following correctly shows the completed venn diagram?
(options a, b, c, d with different venn diagram labelings)

Explanation:

Step1: Analyze Piano Group

Total piano players: 14. Only piano: 5, piano & drums (not guitar): 1, all three: 2. Let \( x \) be piano & guitar (not drums). So \( 5 + 1 + 2 + x = 14 \). Solve: \( x = 14 - 8 = 6 \).

Step2: Analyze Guitar Group

Total guitar players: 18. Only guitar: 7, all three: 2, piano & guitar (not drums): 6. Let \( y \) be guitar & drums (not piano). So \( 7 + 2 + 6 + y = 18 \). Solve: \( y = 18 - 15 = 3 \).

Step3: Analyze Drums Group

Total drums players: 9. Piano & drums (not guitar): 1, all three: 2, guitar & drums (not piano): 3. Let \( z \) be only drums. So \( 1 + 2 + 3 + z = 9 \). Solve: \( z = 9 - 6 = 3 \).

Step4: Match with Options

Check each option. For piano-guitar (not drums): 6, guitar-drums (not piano): 3, only drums: 3. Option b has these values.

Answer:

b