Question:hard

If a certain weight of an alloy of silver and copper is mixed with 3 kg of pure silver, the resulting alloy will have 90% silver by weight. If the same weight of the initial alloy is mixed with 2 kg of another alloy which has 90% silver by weight, the resulting alloy will have 84% silver by weight. Then, the weight of the initial alloy, in kg, is

Updated On: Jan 15, 2026
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The Correct Option is A

Solution and Explanation

Define variables: \( x \) = initial alloy weight (kg), \( y \) = initial alloy silver percentage.

Condition 1: Mixing initial alloy with 3 kg pure silver yields a 90% silver alloy.
Mixture weight: \( x + 3 \) kg.
Silver content: \( \frac{xy}{100} + 3 \) kg.
Equation: \( \frac{\frac{xy}{100} + 3}{x + 3} = 0.9 \)
Simplify to \( \frac{xy}{100} + 3 = 0.9(x + 3) \), then \( \frac{xy}{100} = 0.9x - 0.3 \).
Multiply by 100: \( xy = 90x - 30 \) (Equation 1).

Condition 2: Mixing initial alloy with 2 kg of 90% silver alloy yields an 84% silver alloy.
Mixture weight: \( x + 2 \) kg.
Silver content: \( \frac{xy}{100} + 1.8 \) kg.
Equation: \( \frac{\frac{xy}{100} + 1.8}{x + 2} = 0.84 \)
Simplify to \( \frac{xy}{100} + 1.8 = 0.84(x + 2) \), then \( \frac{xy}{100} = 0.84x - 0.12 \).
Multiply by 100: \( xy = 84x - 12 \) (Equation 2).

Solve the system of equations:
From Equation 1: \( xy = 90x - 30 \)
From Equation 2: \( xy = 84x - 12 \)
Equate the expressions for \( xy \): \( 90x - 30 = 84x - 12 \)
Subtract \( 84x \) from both sides: \( 6x - 30 = -12 \)
Add 30 to both sides: \( 6x = 18 \)
Divide by 6: \( x = 3 \).
The initial alloy weight is 3 kg.
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