13.413 An alloy consists of tin, copper, and zinc. If 20 g is separated from this alloy and melted with 2 g of tin, then in the newly obtained alloy, the mass of copper will be equal to the mass of tin. If, however, 30 g is separated from the original alloy and 9 g of zinc is added, then in this new alloy, the mass of tin will be equal to the mass of zinc. Determine the percentage composition of the original alloy.
Solution. Let $x, y, z$ be the concentrations of copper, tin, and zinc in the alloy. According to the first condition of the problem, $20 x = 20 y + 2$. According to the second condition, $30 y = 30 z + 9$. In the end, we have:

$$
\left\{\begin{array} { l } 
{ 1 0 x = 1 0 y + 1 , } \\
{ 1 0 y = 1 0 z + 3 , } \\
{ x + y + z = 1 }
\end{array} \Rightarrow \left\{\begin{array} { l } 
{ 1 0 x = 1 0 y + 1 , } \\
{ 1 0 y = 1 0 ( 1 - x - y ) + 3 , } \\
{ z = 1 - x - y }
\end{array} \Rightarrow \left\{\begin{array}{l}
x=0.5 \\
y=0.4 \\
z=0.1
\end{array}\right.\right.\right.
$$

Answer: $50\%, 40\%, 10\%$.