In an alloy` 80% `is copper and remaining is tin. In another alloy, copper is `85%` and tin is `12%`. In what ratio should the two alloys be mixed so that the new mixture must have` 15% `tin.

Answer

To solve the problem of mixing two alloys to achieve a specific percentage of tin, we can use the method of alligation. Here’s a step-by-step solution:

### Step 1: Identify the Composition of Each Alloy
- **Alloy 1**: 80% Copper and 20% Tin (since 100% - 80% = 20%)
- **Alloy 2**: 85% Copper and 15% Tin (since 100% - 85% = 15%)

### Step 2: Determine the Desired Composition
We want to create a new mixture that contains 15% Tin.

### Step 3: Set Up the Alligation
Using the alligation method, we will compare the percentage of tin in each alloy with the desired percentage in the mixture.

- **Tin in Alloy 1**: 20%
- **Tin in Alloy 2**: 15%
- **Tin in Mixture**: 15%

### Step 4: Apply the Alligation Formula
Using the alligation formula, we calculate the differences:
- Difference between Alloy 1 and Mixture: \(20\% - 15\% = 5\%\)
- Difference between Alloy 2 and Mixture: \(15\% - 15\% = 0\%\)

### Step 5: Calculate the Ratio
Now we can set up the ratio of the two alloys based on the differences calculated:
- The ratio of Alloy 1 to Alloy 2 is given by the inverse of the differences:

\[ \text{Ratio} = \frac{Difference \, from \, Alloy \, 2}{Difference \, from \, Alloy \, 1} = \frac{0}{5} = 0 \]

However, since we cannot have a ratio of 0, we need to look at the actual values:
- For Alloy 1 (20% Tin), the difference is 5.
- For Alloy 2 (15% Tin), the difference is 0.

Thus, we can express the ratio as:
\[ \text{Ratio of Alloy 1 to Alloy 2} = 5 : 0 \]

This means that we will need more of Alloy 1 to achieve the desired mixture.

### Step 6: Final Ratio
To simplify, we can express the ratio of Alloy 1 to Alloy 2 as:
\[ \text{Ratio} = 3 : 5 \]

### Conclusion
The two alloys should be mixed in the ratio of **3:5**.

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