Alloy A contains copper and zinc in the ratio of `5:2` and alloy B contains copper and zinc in the ratio of `1 :3`. A and B are taken in the ratio of 9:8 and melted to form a new alloy. The percentage of zinc in the new alloy is closest to:

Options

A. 46.9
B. 53.86
C. 48.73
D. 50.42

Answer

To solve the problem, we need to find the percentage of zinc in the new alloy formed by melting alloys A and B in the given ratio. Let's break it down step by step.

### Step 1: Determine the composition of Alloy A
Alloy A contains copper and zinc in the ratio of 5:2.
Let’s assume we have 7 parts of Alloy A (5 parts copper + 2 parts zinc).

- Copper in Alloy A = \( \frac{5}{7} \) of the total weight
- Zinc in Alloy A = \( \frac{2}{7} \) of the total weight

### Step 2: Determine the composition of Alloy B
Alloy B contains copper and zinc in the ratio of 1:3.
Let’s assume we have 4 parts of Alloy B (1 part copper + 3 parts zinc).

- Copper in Alloy B = \( \frac{1}{4} \) of the total weight
- Zinc in Alloy B = \( \frac{3}{4} \) of the total weight

### Step 3: Determine the total weight of Alloys A and B
Alloy A and Alloy B are mixed in the ratio of 9:8.
Let’s assume we take 9 kg of Alloy A and 8 kg of Alloy B.

### Step 4: Calculate the amounts of copper and zinc in each alloy
**For Alloy A (9 kg):**
- Copper in Alloy A = \( 9 \times \frac{5}{7} = \frac{45}{7} \) kg
- Zinc in Alloy A = \( 9 \times \frac{2}{7} = \frac{18}{7} \) kg

**For Alloy B (8 kg):**
- Copper in Alloy B = \( 8 \times \frac{1}{4} = 2 \) kg
- Zinc in Alloy B = \( 8 \times \frac{3}{4} = 6 \) kg

### Step 5: Calculate the total amounts of copper and zinc in the new alloy
**Total Copper:**
- Total Copper = Copper from Alloy A + Copper from Alloy B
- Total Copper = \( \frac{45}{7} + 2 = \frac{45}{7} + \frac{14}{7} = \frac{59}{7} \) kg

**Total Zinc:**
- Total Zinc = Zinc from Alloy A + Zinc from Alloy B
- Total Zinc = \( \frac{18}{7} + 6 = \frac{18}{7} + \frac{42}{7} = \frac{60}{7} \) kg

### Step 6: Calculate the total weight of the new alloy
Total weight of the new alloy = Weight of Alloy A + Weight of Alloy B
- Total weight = 9 kg + 8 kg = 17 kg

### Step 7: Calculate the percentage of zinc in the new alloy
Percentage of Zinc = \( \left( \frac{\text{Total Zinc}}{\text{Total Weight}} \right) \times 100 \)
- Percentage of Zinc = \( \left( \frac{\frac{60}{7}}{17} \right) \times 100 \)

### Step 8: Simplify the expression
- Percentage of Zinc = \( \left( \frac{60}{7 \times 17} \right) \times 100 = \left( \frac{60}{119} \right) \times 100 \approx 50.42\% \)

### Final Answer
The percentage of zinc in the new alloy is approximately **50.42%**.

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