Question

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

• A. 3
• B. 2.5
• C. 4
• D. 3.5

Answer 1

Answer: (A)

3

Answer 2

Let the initial weight of the alloy be $x$ kg.
Let the percentage of silver in the initial alloy be $y$ (in decimal form).

Case 1 : Mixing with pure silver
When the alloy is mixed with 3 kg of pure silver, the resulting alloy will have 90% silver by weight.
Silver from the initial alloy = $xy$ kg.
Silver from 3 kg pure silver = 3 kg.
Total silver in the mixture = $xy + 3$ kg.

Total weight of the mixture = $x + 3$ kg.

Percentage of silver = $\frac{xy + 3}{x + 3} = 0.9$

From this, $xy + 3 = 0.9(x + 3)$
Equation (1): $xy = 0.9x + 0.3$

Case 2 : Mixing with 90% silver alloy
When 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.
Silver from the initial alloy = $xy$ kg.
Silver from 2 kg of 90% silver alloy = 1.8 kg.
Total silver in the mixture = $xy + 1.8$ kg.
Total weight of the mixture = $x + 2$ kg.

Percentage of silver =$\frac{xy + 1.8}{x + 2} = 0.84$

From this, $xy + 1.8 = 0.84(x + 2)$

Equation (2): $xy = 0.84x + 0.32$

Subtracting (2) from (1):

$0.06x = -0.02$ or $x = -\frac{0.02}{0.06}$

or $x = \frac{-1}{3 }=3$