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

The Correct Option is A

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  \( x = \frac{-1}{3 }=3\)

Answer 2

Suppose, alloy contains x Kg of silver and y kg of copper.
According to the question,
\(\frac {x+3}{x+y+3} = \frac {9}{10}\)
\(⇒ 10x+30 =9x+9y+27\)
\(⇒ 9y-x=3\)     ……. \((1)\)
Silver in 2^nd alloy \(= 2 \times 0.9 =1.8\)
\(\frac {x+1.8}{x+y+2} = \frac {21}{25}\)
\(⇒ 21y-4x =3\)     ……. \((2)\)
On solving eq (1) and (2), 
\(y= 0.6\)
and \(x =2.4\)
Then,
\(x+y=2.4+0.6\)
\(x+y=3\)

So, the correct option is (A): \(3\)