## Task 4 - 290614

Of the 25 students in a class, exactly 20 belong to a sports group. Exactly 12 students in this class participate in the Math Club. Exactly 3 students in this class belong neither to a sports group nor to the Math Club.

Show how these data can be used to determine unique numerical answers to the following questions! Provide these answers!

a) How many students in this class belong to the Math Club but not to a sports group?

b) How many students in this class belong to a sports group but not to the Math Club?

c) How many students in this class belong to both the Math Club and a sports group?
From the first statement of the problem text, it follows that $25-20=5$:

Exactly 5 students in the class do not belong to a sports group. Among these, the 3 students mentioned in the third statement of the problem text, who also do not belong to the mathematics club, must be included. Therefore, because $5 - 3 = 2$, the answer to a) is: In total, 2 students in the class belong to the mathematics club but not to a sports group.

These 2 students must be among the 12 students mentioned in the second statement of the problem text.

Because $12 - 2 = 10$, it follows that the answer to c) is: In total, 10 students in the class belong to both the mathematics club and a sports group. From this and the first statement of the problem text, it follows that $20 - 10 = 10$ as the answer to b):

In total, 10 students in the class belong to a sports group but not to the mathematics club.

Note: To verify, the numbers can be entered into the diagram in Figure, and thus confirmed through $10 + 10 + 2 + 3 = 25, 10 + 10 = 20, 10 + 2 = 12$ that the statements in the problem text are satisfied.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0268.jpg?height=300&width=552&top_left_y=1706&top_left_x=752)

Solutions of the 1st Round 1989 adopted from [5]

### 3.30.2 2nd Round 1989, Class 6