2. A fraction in its simplest form was written on the board. Petya decreased its numerator by 1 and its denominator by 2. Vasya, on the other hand, increased the numerator by 1 and left the denominator unchanged. It turned out that the boys ended up with the same value. What exactly could their result be?
Answer. 1.

Solution. Let the fraction $\frac{a}{b}$ have been written. Then Petya obtained $\frac{a-1}{b-2}$, and Vasya obtained $\frac{a+1}{b}$. Since they got the same result, $\frac{a-1}{b-2}=\frac{a+1}{b}$, from which $b-a=1$. Therefore, the original fraction had the form $\frac{a}{a+1}$. And Petya obtained the fraction $\frac{a-1}{a-1}$, and Vasya obtained $\frac{a+1}{a+1}$, i.e., the result of both Petya and Vasya is 1.

Comment.

Since the problem states that Petya and Vasya obtained a certain value, it is not necessary to check that the denominator is not zero.

## Grading criteria.

- Correct solution - 7 points.
- It is found that the original fraction had the form $\frac{a}{a+1}$ (or an equivalent form), but no further conclusions about the final value are made - 3 points.
- The solution is provided with a specific example (for example, it is shown that for the fraction $\frac{2}{3}$ the condition of the problem is satisfied) - 2 points.
- Only the correct answer is given - 1 point.