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Question

Difficulty: MediumMixture and Concentration

An jeweler creates a custom silver alloy by melting together two available alloys. Alloy X is 40%40\% silver by weight, and Alloy Y is 70%70\% silver by weight. If the jeweler uses 1515 grams more of Alloy Y than Alloy X to produce a final alloy mixture that is 60%60\% silver by weight, what is the total weight, in grams, of the final alloy mixture?

  1. A
    1515 grams
  2. B
    3030 grams
  3. 4545 gramsAnswer
  4. D
    6060 grams
  5. E
    7575 grams

Answer

4545 grams
The total weight of the mixture is 4545 grams. Letting xx represent the grams of Alloy X, the weight of Alloy Y is x+15x + 15. Equating the pure silver content gives 0.40x+0.70(x+15)=0.60(2x+15)0.40x + 0.70(x + 15) = 0.60(2x + 15). Solving yields x=15x = 15 grams for Alloy X and 3030 grams for Alloy Y, making the total final mixture weight 15+30=4515 + 30 = 45 grams.

Step-by-Step Solution

1
Define variables for the weights of the alloys.
Let the weight of Alloy X be xx grams. Since Alloy Y has 1515 grams more than Alloy X, the weight of Alloy Y is x+15x + 15 grams. The total weight of the mixture is x+(x+15)=2x+15x + (x + 15) = 2x + 15 grams.
Establishing explicit algebraic terms for component and total weights is essential for setting up the mixture equation.
2
Set up the equation for total silver content.
0.40x+0.70(x+15)=0.60(2x+15)0.40x + 0.70(x + 15) = 0.60(2x + 15)
The total amount of pure silver from Alloy X and Alloy Y combined must equal the amount of pure silver in the final mixture.
3
Solve the algebraic equation for xx.
0.40x+0.70x+10.5=1.20x+9⇒1.10x+10.5=1.20x+9⇒0.10x=1.5⇒x=150.40x + 0.70x + 10.5 = 1.20x + 9 \Rightarrow 1.10x + 10.5 = 1.20x + 9 \Rightarrow 0.10x = 1.5 \Rightarrow x = 15.
Isolating xx determines the weight of Alloy X used.
4
Calculate the total weight of the final alloy mixture.
Total weight = 2x+15=2(15)+15=452x + 15 = 2(15) + 15 = 45 grams.
The question asks for the total weight of the mixture, requiring substitution of x=15x = 15 back into the total weight expression.

Key Concept

Weighted average and concentration conservation in mixtures
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Topics

Mixture and Concentration

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