## 5. Music School

In a certain music school, each student plays exactly one instrument: piano, guitar, trumpet, or violin. Half of all students play the piano, and half of the half of all students play the guitar. The number of students who play the trumpet is equal to a quarter of the number of students who play the guitar, and 72 students play the violin. How many students are there in the music school?

## Result: $\quad 384$
Solution.

The number of students can be represented by a diagram where the entire square represents all the students in the school.

Half of the students who play the piano are marked in red.

![](https://cdn.mathpix.com/cropped/2024_05_30_44689c7bad1ce05a818cg-3.jpg?height=227&width=227&top_left_y=2315&top_left_x=1342)

piano

guitar

trumpet

violin

Those who play the guitar are marked in green (this is half of the students who do not play the piano).

The remaining students play the trumpet or violin. The total number of these students is the same as those who play the guitar.

Since the number of students who play the trumpet is four times less than those who play the guitar, we can mark the corresponding part of the square in blue. The remaining students, who play the violin, are marked in yellow.

Now it is clear that the number of students who play the violin is three times more than those who play the trumpet.

72 students play the trumpet, which is $72: 3=24$ students, and the number of students who play the trumpet or violin is $24+72=96$. The same number of students play the guitar. Twice as many students play the piano, which is 192 students. The music school has a total of $2 \cdot 192=384$ students.

We could have arrived at this result by multiplying $16 \cdot 24=384$ because each small square represents 24 students.