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 Analysis
Let the quantity of alloy X be \(x\) units and the quantity of alloy Y be \(y\) units.
Alloy X contains only Copper (Cu) and Zinc (Zn).
Alloy Y contains 80% Cu, 4% Zn, and 16% Tin (Sn).
The fused mass (mixture of X and Y) contains 74% Cu, 16% Zn, and 10% Sn.
Step 1: Determine the Ratio of Alloys X and Y using Tin Percentage
Tin (Sn) is only present in alloy Y.
- Amount of Tin in the fused mass = 10% of \((x + y)\) = \(0.10(x + y)\).
- Amount of Tin in alloy Y = 16% of \(y\) = \(0.16y\).
- Since the Tin in the fused mass comes only from Y, we have: \(0.10(x + y) = 0.16y\)
- Expanding the equation: \(0.10x + 0.10y = 0.16y\)
- Rearranging to find the ratio \(x:y\): \(0.10x = 0.16y - 0.10y\) \(0.10x = 0.06y\) \(\frac{x}{y} = \frac{0.06}{0.10} = \frac{6}{10} = \frac{3}{5}\)
Let \(x = 3k\) and \(y = 5k\) for some constant \(k\). The total mass is \(x + y = 3k + 5k = 8k\).
Step 2: Calculate the Percentage of Zinc in Alloy X
Let the percentage of Zinc in alloy X be \(z_X\%\).
- Amount of Zinc in X = \(z_X\%\) of \(x\) = \(\frac{z_X}{100} \times 3k\).
- Amount of Zinc in Y = 4% of \(y\) = \(0.04 \times 5k = 0.20k\).
- Total amount of Zinc in the fused mass = 16% of \((x+y)\) = \(0.16 \times 8k = 1.28k\).
- The total Zinc is the sum of Zinc from X and Y: \(\text{Total Zinc} = (\text{Zinc in X}) + (\text{Zinc in Y})\) \(1.28k = \left(\frac{z_X}{100} \times 3k\right) + 0.20k\)
- Divide the entire equation by \(k\) (assuming \(k \neq 0\)): \(1.28 = \frac{3z_X}{100} + 0.20\)
- Solve for \(z_X\): \(1.28 - 0.20 = \frac{3z_X}{100}\) \(1.08 = \frac{3z_X}{100}\) \(z_X = \frac{1.08 \times 100}{3}\) \(z_X = \frac{108}{3}\) \(z_X = 36\)
Therefore, the percentage of zinc in alloy X is 36%.
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