In two alloys, copper and zinc are related in the ra tio of 4:1 and 1:3. 10 kg of 1st alloy, 16 kg of 2nd alloy and some of pure copper are melted together. An alloy was obtained in which the ratio of copper to zinc was 3:2. Find the weight of the new alloy?

Options

A. 35 kg.
B. 45 kg.
C. 40 kg.
D. 50 kg.

Answer

To solve the problem, we need to find the weight of the new alloy formed by melting two alloys and some pure copper. Let's break it down step by step.

### Step 1: Determine the amount of copper and zinc in the first alloy.
The first alloy has a copper to zinc ratio of 4:1. This means that for every 4 parts of copper, there is 1 part of zinc.

- Total weight of the first alloy = 10 kg
- Weight of copper in the first alloy = \( \frac{4}{4+1} \times 10 \) kg
- Weight of zinc in the first alloy = \( \frac{1}{4+1} \times 10 \) kg

Calculating these:
- Weight of copper = \( \frac{4}{5} \times 10 = 8 \) kg
- Weight of zinc = \( \frac{1}{5} \times 10 = 2 \) kg

### Step 2: Determine the amount of copper and zinc in the second alloy.
The second alloy has a copper to zinc ratio of 1:3. This means that for every 1 part of copper, there are 3 parts of zinc.

- Total weight of the second alloy = 16 kg
- Weight of copper in the second alloy = \( \frac{1}{1+3} \times 16 \) kg
- Weight of zinc in the second alloy = \( \frac{3}{1+3} \times 16 \) kg

Calculating these:
- Weight of copper = \( \frac{1}{4} \times 16 = 4 \) kg
- Weight of zinc = \( \frac{3}{4} \times 16 = 12 \) kg

### Step 3: Calculate the total amount of copper and zinc before adding pure copper.
Now, we can sum up the weights of copper and zinc from both alloys:
- Total weight of copper = Weight from first alloy + Weight from second alloy + Weight of pure copper (let's denote this as \( x \))
- Total weight of zinc = Weight from first alloy + Weight from second alloy

Calculating these:
- Total weight of copper = \( 8 + 4 + x = 12 + x \) kg
- Total weight of zinc = \( 2 + 12 = 14 \) kg

### Step 4: Set up the equation based on the new ratio.
The new alloy has a copper to zinc ratio of 3:2. This means:
\[ \frac{\text{Total weight of copper}}{\text{Total weight of zinc}} = \frac{3}{2} \]
Substituting the total weights we found:
\[ \frac{12 + x}{14} = \frac{3}{2} \]

### Step 5: Cross-multiply to solve for \( x \).
Cross-multiplying gives:
\[ 2(12 + x) = 3(14) \]
Expanding both sides:
\[ 24 + 2x = 42 \]

### Step 6: Solve for \( x \).
Rearranging the equation:
\[ 2x = 42 - 24 \]
\[ 2x = 18 \]
\[ x = 9 \text{ kg} \]

### Step 7: Calculate the total weight of the new alloy.
Now we can find the total weight of the new alloy:
\[ \text{Total weight of new alloy} = \text{Weight of first alloy} + \text{Weight of second alloy} + \text{Weight of pure copper} \]
\[ = 10 + 16 + 9 = 35 \text{ kg} \]

### Final Answer:
The weight of the new alloy is **35 kg**.

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