The seventh grades of our school competed in collecting caps from PET bottles. Class A collected half of what classes B and C collected together, class B collected a third of what classes A and C collected together, and class C collected 150 caps.

Determine how many caps these three classes collected in total.

(M. Volfová)
Class A collected half of what classes B and C collected together; this means that class A collected a third of the total number of caps. Class B collected a third of what classes A and C collected together; this means that class B collected a quarter of the total number of caps. Therefore, classes A and B together collected

$$
\frac{1}{3}+\frac{1}{4}=\frac{4}{12}+\frac{3}{12}=\frac{7}{12}
$$

of the total number of caps.

Class C thus collected $\frac{5}{12}$ of the total number of caps, which amounted to 150 pieces. This implies that one twelfth of the total collection was $150: 5=30$ caps. Therefore, all classes together collected $12 \cdot 30=360$ caps.

A helpful graphical representation is shown in the following image:

![](https://cdn.mathpix.com/cropped/2024_04_17_7af2ed77a4c56d0b6466g-1.jpg?height=368&width=740&top_left_y=1986&top_left_x=661)

Scoring suggestion. 2 points for recognizing that class A collected a third of the total number of caps; 1 point for a similar recognition for class B; 2 points for recognizing that class C collected $\frac{5}{12}$ of the total number; 1 point for solving the problem.