13.389 From two pieces of alloy of the same mass but with different percentage content of copper, pieces of equal mass were cut off. Each of the cut pieces was melted with the remainder of the other piece, after which the percentage content of copper in both pieces became the same. How many times smaller is the cut piece than the whole?
Solution. Let $y$ be the mass of each alloy piece, $x$ be the mass of the cut-off piece, $p$ and $q$ be the concentration of copper in the first and second pieces of the alloy. Then $p x+q(y-x)$ and $q x+p(y-x)$ are the amounts of copper in the new alloys, respectively. According to the problem,

$$
\frac{p x+q(y-x)}{y}=\frac{q x+p(y-x)}{y} \Rightarrow(p-q) x=(y-x)(p-q) \Rightarrow x=y-x \Rightarrow y=2 x
$$

Answer: 2 times.