A and B are two alloys of copper and tin prepared by mixing the respective metals in the ratio of 5 : 3 and 5 : 11 respectively. If the alloys A and B are mixed to from a third alloy C with an equal proportion of copper and tin, what is the ratio of alloys A and B in the new alloy C?

Options

A. 3:5
B. 4:5
C. 3:2
D. 2:3

Answer

To solve the problem, we need to determine the ratio of alloys A and B when mixed to form a new alloy C that has an equal proportion of copper and tin. Let's break this down step by step.

### Step 1: Understand the Composition of Alloys A and B

- **Alloy A** is made of copper and tin in the ratio of 5:3.
- Total parts = 5 + 3 = 8
- Copper in A = \( \frac{5}{8} \)
- Tin in A = \( \frac{3}{8} \)

- **Alloy B** is made of copper and tin in the ratio of 5:11.
- Total parts = 5 + 11 = 16
- Copper in B = \( \frac{5}{16} \)
- Tin in B = \( \frac{11}{16} \)

### Step 2: Determine the Composition of Alloy C

Alloy C is required to have an equal proportion of copper and tin, which means:
- Copper in C = Tin in C = \( \frac{1}{2} \)

### Step 3: Set Up the Allegation Method

We can use the allegation method to find the ratio in which alloys A and B must be mixed to achieve the desired composition in alloy C.

1. **Calculate the difference between the copper proportions:**
- Copper in A = \( \frac{5}{8} = 0.625 \)
- Copper in B = \( \frac{5}{16} = 0.3125 \)
- Copper in C = \( 0.5 \)

Using the allegation method:
- Difference between A and C: \( 0.625 - 0.5 = 0.125 \)
- Difference between B and C: \( 0.5 - 0.3125 = 0.1875 \)

2. **Set up the ratio using the differences:**
- The ratio of A to B is given by the differences calculated:
- Ratio of A to B = \( 0.1875 : 0.125 \)

### Step 4: Simplify the Ratio

To simplify the ratio:
- Convert the decimals to fractions:
- \( 0.1875 = \frac{3}{16} \)
- \( 0.125 = \frac{1}{8} \)

Now, we can express the ratio:
- Ratio of A to B = \( \frac{3/16}{1/8} = \frac{3}{16} \times \frac{8}{1} = \frac{3 \times 8}{16} = \frac{24}{16} = \frac{3}{2} \)

### Final Ratio

Thus, the ratio of alloys A and B in the new alloy C is **3:2**.

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