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%.