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Difficulty: Very hardRatios, Rates, and Proportions

A metal alloy is composed of copper, zinc, and nickel. In the original alloy, the ratio of the weight of copper to zinc is 3:43:4, and the ratio of the weight of zinc to nickel is 4:54:5. A metallurgist melts this 240240-gram alloy and adds a mixture of pure copper and pure nickel. The weight of the copper added is twice the weight of the nickel added. In the resulting alloy, the weight of copper is equal to the weight of nickel. What is the ratio of the weight of zinc to the total weight of the new alloy?

  1. A
    415\frac{4}{15}
  2. B
    13\frac{1}{3}
  3. 29\frac{2}{9}Answer
  4. D
    27\frac{2}{7}
  5. E
    433\frac{4}{33}

Answer

The correct ratio of the weight of zinc to the total weight of the new alloy is 29\frac{2}{9}.
The correct answer is 29\frac{2}{9}. To find this, we first establish the joint ratio of the metals as Copper : Zinc : Nickel = 3:4:53 : 4 : 5. For a 240240-gram alloy, this gives 6060 g of copper, 8080 g of zinc, and 100100 g of nickel. Adding xx grams of nickel and 2x2x grams of copper yields a new copper weight of 60+2x60 + 2x and a new nickel weight of 100+x100 + x. Since these weights are equal in the new alloy, 60+2x=100+x  ⟹  x=4060 + 2x = 100 + x \implies x = 40. The new total weight is 240+3(40)=360240 + 3(40) = 360 grams. The weight of zinc remains 8080 grams, so the ratio of zinc to the total weight is 80/360=2980/360 = \frac{2}{9}.

Step-by-Step Solution

1
Express the ratio of the three metals in the original alloy.
Copper : Zinc : Nickel = 3:4:53 : 4 : 5
Since Copper : Zinc = 3:43:4 and Zinc : Nickel = 4:54:5, the zinc terms are already aligned, allowing us to combine them into a single three-part ratio.
2
Calculate the initial weights of copper, zinc, and nickel in the 240240-gram alloy.
Copper = 6060 grams, Zinc = 8080 grams, Nickel = 100100 grams
The sum of the ratio parts is 3+4+5=123 + 4 + 5 = 12. Each part represents 240 grams/12=20240 \text{ grams} / 12 = 20 grams. Multiplying each part by 20 gives the initial weights.
3
Define variables for the added metals and set up the equation for the new weights.
New Copper = 60+2x60 + 2x, New Nickel = 100+x100 + x
Let xx represent the weight of nickel added in grams. Since the weight of the copper added is twice that of the nickel added, 2x2x grams of copper are added.
4
Solve for the value of xx using the condition that the final weights of copper and nickel are equal.
x=40x = 40
Setting the expressions equal gives 60+2x=100+x60 + 2x = 100 + x. Subtracting xx and 6060 from both sides yields x=40x = 40 grams.
5
Determine the new total weight of the alloy and calculate the final ratio of zinc to the total weight.
New Total Weight = 360360 grams, Final Ratio = 29\frac{2}{9}
The new total weight is the original weight plus the added metals: 240+3(40)=360240 + 3(40) = 360 grams. Since no zinc was added, the weight of zinc remains 8080 grams. The ratio is 80/360=2980 / 360 = \frac{2}{9}.

Key Concept

Combining ratios and setting up algebraic equations to model changing quantities in mixtures.
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Topics

Ratios, Rates, and Proportions

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