Problema Solution
Problem: There are 60 people in a music class.
20 play the flute, 10 play the violin, 20 play the piano.
10 play both the flute and the piano, and 5 play all three instruments. How many students don't play any instruments?
Answer provided by our tutors
Venn Diagram:
We see that 10 - 5 = 5 people play the flute and the piano (but nothing else)
20 - 15 = 5 people play just the flute only
20 - 15 = 5 people play just the piano only
10 people play just the violin only
10-5 =5 people play the violin and one other instrument only
5 people play all three
Adding those up, we get:
5 + 5 + 5 + 10 + 5 + 5 =
10 + 15 + 10 = 35
Therefore 25 students do not play a single instrument.
We can also arrange these numbers into sets:
Violin only player is the set of 10 students. If we assign each of the 60 students a number (student #1 = 1, student #2 = 2 ... student #60 = 60) then we can write:
Violins Only: [1, 2, 3, 4 ... 9, 10]
Flute Only: [11, 12, 13, 14 ,15]
Piano Only: [16, 17, 18, 19, 20]
Violin & Piano or Violin & Flute: [21, 22, 23, 24, 25]
All Three: [26, 27, 28, 29, 30]
None (don't play an instrument): [31, 32, 33 ... 59, 60]
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