2. Two farmers brought a total of 100 eggs to the market. They had a different number of eggs, but they received the same amount of money for them. The first said to the second: "If I had your eggs, I would get 15 euros for them." The second replied: "If I sold your eggs, I would get $6 \frac{2}{3}$ euros for them." How many eggs did each have?

(7 points)
2. If the first farmer had $x$ eggs, then the second had $100-x$. If the first farmer had $100-x$ eggs, she would have received 15 euros for them. Therefore, she was selling her eggs at $\frac{15}{(100-x)}$ each. If the second farmer had $x$ eggs, she would have received $6 \frac{2}{3}$ euros for them. Therefore, she was selling her eggs at $6 \frac{2}{3}: x=\frac{20}{3 x}$ each. The first farmer received $\frac{x \cdot 15}{(100-x)}=\frac{15 x}{(100-x)}$ for her eggs, and the second received $(100-x) \cdot \frac{20}{3 x}=\frac{20(100-x)}{3 x}$. Since both received the same amount of money, we can write the equation $\frac{15 x}{(100-x)}=\frac{20(100-x)}{3 x}$. When we simplify the equation, we get the quadratic equation $x^{2}+160 x-8000=0$, which has two solutions, namely $x_{1}=40$ and $x_{2}=-200$. Since a negative $x$ is nonsensical in our case, the solution to the problem is that the first farmer had 40 eggs, and the second had 60 eggs.
Usage or notation: the first farmer $x$ eggs, the second $100-x$ ..... 1 point
Price of one egg from the first farmer $\frac{15}{(100-x)}$ and price of one egg from the second farmer $\frac{20}{3 x}$.... ..... 1 point
Writing the equation $\frac{15 x}{(100-x)}=\frac{20(100-x)}{3 x}$ ..... 1 point
Simplifying the equation to $x^{2}+160 x-8000=0$ ..... 1 point
Solutions to the equation $x_{1}=40$ and $x_{2}=-200$...................... ..... 1 point
Excluding the second solution ..... 1 point
Answer: The first farmer had 40 eggs, and the second had 60 eggs ..... 1 point