A sample of a certain alloy has a total mass of 50.0 grams and is 50.0

Choice B is correct. Let $x$ represent the total mass, in grams, of the first piece, and let $y$ represent the total mass, in grams, of the second piece. It's given that the sample has a total mass of $50.0$ grams. Therefore, the equation $x+y=50.0$ represents this situation. It's also given that the sample is $50.0\%$ silicon by mass. Therefore, the total mass of the silicon in the sample is $0.500\left(50.0\right)$, or $25.0$, grams. It's also given that the first piece was $30.0\%$ silicon by mass and the second piece was $80.0\%$ silicon by mass. Therefore, the masses, in grams, of the silicon in the first and second pieces can be represented by the expressions $0.300x$ and $0.800y$, respectively. Since the sample was created by combining the first and second pieces, and the total mass of the silicon in the sample is $25.0$ grams, the equation $0.300x+0.800y=25.0$ represents this situation. Subtracting $y$ from both sides of the equation $x+y=50.0$ yields $x=50.0-y$. Substituting $50.0-y$ for $x$ in the equation $0.300x+0.800y=25.0$ yields $0.300\left(50.0-y\right)+0.800y=25.0$. Distributing $0.300$ on the left-hand side of this equation yields $15.0-0.300y+0.800y=25.0$. Combining like terms on the left-hand side of this equation yields $15.0+0.500y=25.0$. Subtracting $15.0$ from both sides of this equation yields $0.500y=10.0$. Dividing both sides of this equation by $0.500$ yields $y=20.0$. Substituting $20.0$ for $y$ in the expression representing the mass, in grams, of the silicon in the second piece, $0.800y$, yields $0.800\left(20.0\right)$, or $16.0$. Therefore, the mass, in grams, of the silicon in the second piece is $16.0$.

Choice A is incorrect. This is the mass, in grams, of the silicon in the first piece, not the second piece.

Choice C is incorrect. This is the total mass, in grams, of the second piece, not the mass, in grams, of the silicon in the second piece.

Choice D is incorrect. This is the total mass, in grams, of the first piece, not the mass, in grams, of the silicon in the second piece.