A sample of a certain alloy has a total mass of 50.0 grams and is 50.0 % silicon by mass.

Question

A sample of a certain alloy has a total mass of $50.0$ grams and is $50.0 percent sign$ silicon by mass. The sample was created by combining two pieces of different alloys. The first piece was $30.0 percent sign$ silicon by mass and the second piece was $80.0 percent sign$ silicon by mass. What was the mass, in grams, of the silicon in the second piece?

A.

$9.0$

B.

$16.0$

C.

$20.0$

D.

$30.0$

Accepted Answer

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 50.0 , 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 0.300 x and 0.800 y , 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 0.300 x + 0.800 y = 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 0.300 x + 0.800 y = 25.0 yields 0.300 50.0 - y + 0.800 y = 25.0 . Distributing 0.300 on the left-hand side of this equation yields 15.0 - 0.300 y + 0.800 y = 25.0 . Combining like terms on the left-hand side of this equation yields 15.0 + 0.500 y = 25.0 . Subtracting 15.0 from both sides of this equation yields 0.500 y = 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, 0.800 y , yields 0.800 20.0 , 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.

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