Problem 3. A dog and a cat simultaneously grabbed a sausage loaf with their teeth from different sides. If the dog bites off its piece and runs away, the cat will get 300 g more than the dog. If the cat bites off its piece and runs away, the dog will get 500 g more than the cat. How much sausage will be left if both bite off their pieces and run away?

$[2$ points] (A. V. Shapovalov)
Answer: 400 g.

Solution: If two identical dogs grab the sausage from both sides, there will be a piece of 300 g between them. If two identical cats grab the sausage from both sides, there will be a piece of 500 g between them (see the figure).
![](https://cdn.mathpix.com/cropped/2024_05_06_031fac0a0a7981eac22cg-04.jpg?height=164&width=1072&top_left_y=628&top_left_x=188)

It is clear that a dog bites off ( $500-300$ ) $: 2=100$ g more than a cat. Therefore, when the dog and the cat bite off their pieces and run away, 300 + 100 = 400 g of sausage will remain.

Alternatively, we can reason as follows. The dog intends to bite off 300 g less from the sausage than leave. If he bit off 150 g more, leaving 150 g less, he would get exactly half of the entire sausage. Similarly, the cat would get half of the entire sausage if he bit off 250 g more than he intended. In this case, the entire sausage would be eaten, so the remaining weight is 150 + 250 = 400 g.

Comment: The problem can also be solved algebraically (see solutions for seventh-grade problems).