X is an alloy of copper and zinc. Y is an alloy containing 80% of copper, 4% of zinc and 16% of tin. A fused mass of X and Y is found to contain 74% of copper, 16% of zinc and 10% of tin.
Alloy Mixture: Finding Copper Percentage in X
This solution explains how to determine the percentage of copper in alloy X given the compositions of alloy Y and the final mixture of X and Y.
Step 1: Analyze Component Percentages
We are given the compositions:
- Alloy Y: 80% Copper (Cu), 4% Zinc (Zn), 16% Tin (Sn)
- Mixture (X+Y): 74% Cu, 16% Zn, 10% Sn
- Alloy X: Contains only Copper (Cu) and Zinc (Zn)
Step 2: Determine the Ratio of Alloys X and Y
Let the total weight of the mixture be \(W\). Let the weight of alloy X be \(w_X\) and the weight of alloy Y be \(w_Y\). So, \(W = w_X + w_Y\).
We use the percentage of Tin (Sn) to find the ratio, as only alloy Y contains tin.
- Tin in the Mixture must come entirely from Alloy Y.
- Amount of Tin in Mixture = \(10\%\) of \(W\)
- Amount of Tin from Alloy Y = \(16\%\) of \(w_Y\)
- Therefore, \(0.10 \times W = 0.16 \times w_Y\).
- Calculating the ratio \(\frac{W}{w_Y} = \frac{0.16}{0.10} = \frac{16}{10} = \frac{8}{5}\).
- Since \(W = w_X + w_Y\), we have \(w_X + w_Y = \frac{8}{5} w_Y\).
- Solving for \(w_X\): \(w_X = \frac{8}{5} w_Y - w_Y = (\frac{8}{5} - 1) w_Y = \frac{3}{5} w_Y\).
- This gives the ratio of the weights \(\frac{w_X}{w_Y} = \frac{3}{5}\). Alloy X and Alloy Y are mixed in the ratio 3:5 by weight.
Step 3: Calculate Copper Content in the Mixture
Assume we have 3 units of Alloy X and 5 units of Alloy Y, making a total of \(3 + 5 = 8\) units for the mixture.
- Total Copper in the 8-unit mixture = \(74\%\) of \(8 = 0.74 \times 8 = 5.92\) units.
- Copper contributed by 5 units of Alloy Y = \(5 \times (80\% \text{ of 1 unit}) = 5 \times 0.80 = 4\) units.
Step 4: Calculate Copper Percentage in Alloy X
Let \(c_X\) be the percentage of copper in alloy X.
Copper contributed by 3 units of Alloy X = \(3 \times (c_X\% \text{ of 1 unit}) = 3 \times \frac{c_X}{100}\) units.
The total copper in the mixture is the sum of copper from Alloy X and Alloy Y:
Copper from X + Copper from Y = Total Copper in Mixture
\(3 \times \frac{c_X}{100} + 4 = 5.92\)
Step 5: Solve for \(c_X\)
Rearrange the equation to solve for \(c_X\):
\(3 \times \frac{c_X}{100} = 5.92 - 4\)
\(3 \times \frac{c_X}{100} = 1.92\)
\(3 \times c_X = 1.92 \times 100\)
\(3 \times c_X = 192\)
\(c_X = \frac{192}{3}\)
\(c_X = 64\)
Thus, the percentage of copper in alloy X is 64%.
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