In a school, one quarter of the students play only volleyball, one third play only football, 300 practice both sports, and 1/12 play neither.

(a) How many students are there in the school?

(b) How many students play only football?

(c) How many students play football?

(d) How many students practice one of the 2 sports?
(a)

The total number of students in the school is given by the fraction 12/12, which we can graphically represent by a rectangle divided into 12 equal parts.

![](https://cdn.mathpix.com/cropped/2024_05_01_b9d1398cc05f7e479435g-5.jpg?height=251&width=326&top_left_y=1208&top_left_x=1502)

We will denote by V, F, and NE the number of students who play only volleyball, only football, and neither of these sports, respectively. Now we have:

- the $1 / 4$ of students who play only volleyball correspond to 3 squares;
- the $1 / 3$ of students who play only football correspond to 4 squares;
- the 1/12 of students who do not play any of these sports correspond to 1 square.

| $\mathrm{V}$ | $\mathrm{V}$ | $\mathrm{V}$ | $\mathrm{F}$ |
| :---: | :---: | :---: | :---: |
| $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{NE}$ |
|  |  |  |  |

Thus, 4 rectangles remain for the students who do not play volleyball or football, meaning these 300 students correspond to 4/12 = 1/3 of the total number of students in the school. Therefore, the total number of students in the school is

$$
300 \times 3=900
$$

(b) We have that $\frac{1}{3} \cdot 900=300$ is the total number of students who play only football.

(c) In this case, the students who play football are those who play only football plus those who play both football and volleyball, i.e., $300+300=600$.

(d) The total number of students who practice one of the sports is $\frac{11}{12} \cdot 900=825$, since $1 / 12$ of the students do not practice any of the sports.