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's define the variables:
Let \( x \) be the weight of the initial alloy in kg.
Let \( y \) be the percentage of silver in the initial alloy.

The first condition states that mixing the initial alloy with 3 kg of pure silver results in an alloy of 90% silver. The total weight of this mixture is \( x + 3 \) kg, and the silver content is \( \frac{xy}{100} + 3 \) kg.
Setting up the equation based on the given percentage:
\[ \frac{\frac{xy}{100} + 3}{x + 3} = 0.9 \]
Multiplying through by \( x + 3 \):
\[ \frac{xy}{100} + 3 = 0.9(x + 3) \]
Simplify to:
\[ \frac{xy}{100} + 3 = 0.9x + 2.7 \]
Rearranging gives:
\[ \frac{xy}{100} = 0.9x - 0.3 \]
Multiplying through by 100 for simplicity:
\[ xy=90x - 30 \] (Equation 1)

The second condition involves mixing the initial alloy with another 2 kg alloy that's 90% silver. The total weight is \( x + 2 \) kg, and the silver content is \( \frac{xy}{100} + 1.8 \) kg (since 90% of 2 is 1.8 kg).
Set up the equation:
\[ \frac{\frac{xy}{100} + 1.8}{x + 2} = 0.84 \]
Multiplying through by \( x + 2 \):
\[ \frac{xy}{100} + 1.8=0.84(x+2) \]
Simplify to:
\[ \frac{xy}{100} + 1.8=0.84x + 1.68 \]
Rearranging gives:
\[ \frac{xy}{100} = 0.84x - 0.12 \]
Multiplying through by 100:
\[ xy=84x - 12 \] (Equation 2)

Now solve the two equations:
From Equation 1:
\[ xy = 90x - 30 \]
From Equation 2:
\[ xy = 84x - 12 \]
Set them equal since they both equal \( xy \):
\[ 90x - 30=84x - 12 \]
Subtract \( 84x \) from both sides:
\[ 6x - 30=-12 \]
Add 30 to both sides:
\[ 6x=18 \]
Divide by 6:
\[ x=3 \]
The weight of the initial alloy is 3 kg.

Answer 2

Suppose an alloy contains \( x \) kg of silver and \( y \) kg of copper.

Step 1: Using the first ratio condition

According to the problem, \[ \frac{x + 3}{x + y + 3} = \frac{9}{10} \]

Cross-multiply: \[ 10(x + 3) = 9(x + y + 3), \] which simplifies to \[ 10x + 30 = 9x + 9y + 27 \implies 9y - x = 3 \tag{1} \]

Step 2: Using the second ratio condition

Silver in the second alloy = \(2 \times 0.9 = 1.8\) kg

Given, \[ \frac{x + 1.8}{x + y + 2} = \frac{21}{25} \]

Cross-multiplying gives \[ 25(x + 1.8) = 21(x + y + 2) \implies 25x + 45 = 21x + 21y + 42, \] which simplifies to \[ 21y - 4x = 3 \tag{2} \]

Step 3: Solve the system of equations (1) and (2)

From equations: \[ \begin{cases} 9y - x = 3 \\ 21y - 4x = 3 \end{cases} \]

Solving yields: \[ y = 0.6, \quad x = 2.4 \]

Step 4: Find total weight of the alloy

\[ x + y = 2.4 + 0.6 = 3 \]

Answer:

The total weight of the original alloy is 3 kg.

Therefore, the correct option is (A): 3.