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 of these two sports.

(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 at least one of the two sports?
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_666a1aec7e447b0a6cbeg-072.jpg?height=145&width=328&top_left_y=270&top_left_x=1549)

Let V, F, and NE denote the number of students who play only volleyball, only football, and neither of these two sports, respectively. The information given, in terms of the parts of this rectangle, is as follows:

- $1 / 4$ of the students who play only volleyball corresponds to three parts;
- $1 / 3$ of the students who play only football corresponds to four parts;
- $1 / 12$ of the students who do not play either volleyball or football corresponds to one part.

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

(a) There are 4 parts of the rectangle left for the students who play both volleyball and football, i.e., 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) The total number of students who play only football is $\frac{1}{3} \cdot 900 = 300$.

(c) 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 play either football or volleyball.