9.1. There are two ingots of different copper and tin alloys weighing 6 and 12 kg respectively. From each of them, a piece of the same weight was cut off and the first piece was alloyed with the remainder of the second ingot, and the second piece - with the remainder of the first ingot, after which the ratio of copper to tin in the two resulting new ingots turned out to be the same. Find the weight of each of the cut-off pieces.
Answer: 4 kilograms.

Solution: Let the weight of each of the cut pieces be $x$ kg, and the proportions of tin in the first and second ingots be $a \neq b$ respectively. Then, the proportion of tin after remelting in the first ingot will be $\frac{b x + a(6 - x)}{6}$, and in the second ingot $\frac{a x + b(12 - x)}{12}$, and they are equal according to the condition. Solving the obtained equation, we find $x = 4$ for all $a \neq b$.

Grading criteria. Correct answer with verification for arbitrary $a \neq b: 1$ point. Analysis of a specific case: 0 points.