Two alloys are both made up of copper and tin. The ratio of copper and tin in the first alloy is 1:3 and in the second alloy is 2:5. In what ratio should the two alloys be mixed to obtain a new alloy in which the ratio of tin and copper be 8:3?

Options

A. `3:5`
B. `4:7`
C. `3:8`
D. `5:11`

Answer

To solve the problem, we need to determine the ratio in which the two alloys should be mixed to achieve a new alloy with a specific ratio of tin to copper. Let's break down the solution step by step.

### Step 1: Understand the Ratios of the Alloys
- **Alloy 1** has a ratio of copper to tin as 1:3.
- **Alloy 2** has a ratio of copper to tin as 2:5.

From these ratios, we can determine the fraction of copper and tin in each alloy.

**Alloy 1:**
- Total parts = 1 + 3 = 4
- Copper in Alloy 1 = \( \frac{1}{4} \)
- Tin in Alloy 1 = \( \frac{3}{4} \)

**Alloy 2:**
- Total parts = 2 + 5 = 7
- Copper in Alloy 2 = \( \frac{2}{7} \)
- Tin in Alloy 2 = \( \frac{5}{7} \)

### Step 2: Set Up the Desired Ratio
We want to mix these two alloys to create a new alloy with a ratio of tin to copper as 8:3. This means:
- Total parts = 8 + 3 = 11
- Fraction of tin = \( \frac{8}{11} \)
- Fraction of copper = \( \frac{3}{11} \)

### Step 3: Use Alligation to Find the Mixing Ratio
We can use the alligation method to find the ratio in which the two alloys should be mixed.

1. **Calculate the effective fractions:**
- Fraction of tin in Alloy 1 = \( \frac{3}{4} \)
- Fraction of copper in Alloy 1 = \( \frac{1}{4} \)
- Fraction of tin in Alloy 2 = \( \frac{5}{7} \)
- Fraction of copper in Alloy 2 = \( \frac{2}{7} \)

2. **Calculate the required fractions:**
- Required fraction of tin = \( \frac{8}{11} \)
- Required fraction of copper = \( \frac{3}{11} \)

3. **Set up the alligation:**
- For Alloy 1:
- Tin: \( \frac{3}{4} \) - \( \frac{8}{11} \)
- Copper: \( \frac{1}{4} \) - \( \frac{3}{11} \)

- For Alloy 2:
- Tin: \( \frac{5}{7} \) - \( \frac{8}{11} \)
- Copper: \( \frac{2}{7} \) - \( \frac{3}{11} \)

4. **Calculate the differences:**
- For Alloy 1 (Tin):
\[ \frac{3}{4} - \frac{8}{11} = \frac{33 - 32}{44} = \frac{1}{44} \]
- For Alloy 2 (Copper):
\[ \frac{5}{7} - \frac{8}{11} = \frac{55 - 56}{77} = -\frac{1}{77} \]

### Step 4: Find the Mixing Ratio
The ratio of the two alloys is given by the inverses of the differences calculated:
\[ \text{Ratio} = \frac{\frac{1}{44}}{-\frac{1}{77}} = \frac{77}{44} = \frac{7}{4} \]

### Final Answer
The two alloys should be mixed in the ratio of **7:4**.

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