Two alloys composed of gold and silver together weight 20 kg. One lump contains 75% gold and 31.25 gm per kg silver. Another alloy contains 85% gold and 30 gm per kg silver. The total quantity of silver in two lumps is 617.5 gm. If the two lumps are melted and formed into one, what percentage of gold will it contain ?

Options

A. A)50%
B. B)89%
C. C)78%
D. D)67%

Answer

To solve the problem step by step, we need to analyze the two alloys and their compositions. Here's how we can do it:

### Step 1: Understand the Composition of Each Alloy
- **Alloy 1**: Contains 75% gold and 31.25 gm of silver per kg.
- **Alloy 2**: Contains 85% gold and 30 gm of silver per kg.

### Step 2: Set Up the Total Weight of Alloys
The total weight of both alloys combined is 20 kg (or 20,000 grams).

### Step 3: Calculate the Total Silver Content
We know the total quantity of silver in both alloys is 617.5 gm.

### Step 4: Calculate the Silver Content per kg
- For Alloy 1: Silver content = 31.25 gm/kg
- For Alloy 2: Silver content = 30 gm/kg

### Step 5: Set Up the Equation for Silver
Let the weight of Alloy 1 be \( x \) kg and the weight of Alloy 2 be \( (20 - x) \) kg. The total silver from both alloys can be represented as:
\[ 31.25x + 30(20 - x) = 617.5 \]

### Step 6: Solve for \( x \)
Expanding the equation:
\[ 31.25x + 600 - 30x = 617.5 \]
\[ 1.25x + 600 = 617.5 \]
\[ 1.25x = 617.5 - 600 \]
\[ 1.25x = 17.5 \]
\[ x = \frac{17.5}{1.25} = 14 \]
So, Alloy 1 weighs 14 kg and Alloy 2 weighs \( 20 - 14 = 6 \) kg.

### Step 7: Calculate the Gold Content in Each Alloy
- **Gold in Alloy 1**:
\[ \text{Gold} = 75\% \text{ of } 14 \text{ kg} = 0.75 \times 14 = 10.5 \text{ kg} \]
- **Gold in Alloy 2**:
\[ \text{Gold} = 85\% \text{ of } 6 \text{ kg} = 0.85 \times 6 = 5.1 \text{ kg} \]

### Step 8: Total Gold in the Mixture
The total gold in the combined mixture is:
\[ 10.5 \text{ kg} + 5.1 \text{ kg} = 15.6 \text{ kg} \]

### Step 9: Calculate the Percentage of Gold in the Final Mixture
The total weight of the final mixture is 20 kg. Therefore, the percentage of gold is:
\[ \text{Percentage of Gold} = \left( \frac{15.6 \text{ kg}}{20 \text{ kg}} \right) \times 100 = 78\% \]

### Final Answer
The percentage of gold in the final mixture is **78%**.

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