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 form a third alloy C with an equal proportional of copper and tin, what is ther 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 step by step, we need to find the ratio of alloys A and B that can be mixed to form alloy C, which has equal proportions of copper and tin.

### Step 1: Understand the Ratios of Alloys A and B
- Alloy A is made of copper and tin in the ratio of 5:3.
- Alloy B is made of copper and tin in the ratio of 5:11.

### Step 2: Calculate the Fraction of Copper in Each Alloy
For Alloy A:
- Total parts = 5 (copper) + 3 (tin) = 8
- Fraction of copper in Alloy A = \( \frac{5}{8} \)

For Alloy B:
- Total parts = 5 (copper) + 11 (tin) = 16
- Fraction of copper in Alloy B = \( \frac{5}{16} \)

### Step 3: Calculate the Fraction of Copper in Alloy C
Alloy C has equal proportions of copper and tin, which means:
- Fraction of copper in Alloy C = \( \frac{1}{2} \)

### Step 4: Set Up the Allegation Method
We will use the allegation method to find the ratio of alloys A and B.

- The fraction of copper in Alloy A = \( \frac{5}{8} \)
- The fraction of copper in Alloy B = \( \frac{5}{16} \)
- The fraction of copper in Alloy C = \( \frac{1}{2} \)

### Step 5: Calculate the Differences for Allegation
1. Calculate \( \frac{1}{2} - \frac{5}{16} \):
- Convert \( \frac{1}{2} \) to a fraction with a denominator of 16: \( \frac{1}{2} = \frac{8}{16} \)
- So, \( \frac{1}{2} - \frac{5}{16} = \frac{8}{16} - \frac{5}{16} = \frac{3}{16} \)

2. Calculate \( \frac{5}{8} - \frac{1}{2} \):
- Convert \( \frac{1}{2} \) to a fraction with a denominator of 8: \( \frac{1}{2} = \frac{4}{8} \)
- So, \( \frac{5}{8} - \frac{1}{2} = \frac{5}{8} - \frac{4}{8} = \frac{1}{8} \)

### Step 6: Set Up the Ratio
Now we can set up the ratio of alloys A to B:
- The ratio is given by the differences calculated:
- Ratio of A to B = \( \frac{3/16}{1/8} \)

### Step 7: Simplify the Ratio
To simplify \( \frac{3/16}{1/8} \):
- Multiply by the reciprocal: \( \frac{3}{16} \times \frac{8}{1} = \frac{3 \times 8}{16} = \frac{24}{16} = \frac{3}{2} \)

### Final Ratio
Thus, the ratio of alloys A to B in the new alloy C is:
- **Ratio of A to B = 3:2**