3. Sixteen boys gathered for fishing. It is known that every boy who put on boots also put on a cap. Without boots, there were 10 boys, and without a cap - two. Which boys are more and by how many: those who wore a cap but no boots, or those who put on boots? Be sure to explain your answer.
Answer. Those who were in caps but without boots were 2 more than those who were in boots.

Solution. Out of 16 boys, 10 were without boots, which means 6 were in boots. Two were without caps, so 14 were in caps. Since everyone who wore boots also wore a cap, out of the 14 who wore caps, 6 also wore boots, and the remaining 8 did not wear boots. Therefore, those who were in caps and without boots (8 people) are 2 more than those who were in boots (6 people).

This reasoning can be conducted based on the diagram:

![](https://cdn.mathpix.com/cropped/2024_05_06_843bb5844bda57b2e45dg-2.jpg?height=609&width=945&top_left_y=969&top_left_x=561)

## Grading Criteria.

- Complete and justified solution - 7 points.
- Correct reasoning was conducted, but the final step was not taken (the number of those in caps and without boots and those in boots were given separately), 6 points.
- A correct diagram was drawn or correct reasoning was started, but the solution was not completed - 2 points.
- Correct answer without explanation (but with verification that everything matches) 2 points.
- Only the correct answer - 1 point.