Mixing Gold and Silver Alloys to Achieve a Desired Ratio

The process of mixing two or more alloys to achieve a specific gold to silver ratio is a common practice in metallurgy. This article explores the mathematical approach to determine the ideal mixing ratio of two alloys with different gold to silver ratios to obtain a new alloy with a desired ratio.

Introduction to the Problem

Consider two alloys, A and B, containing gold and silver in the ratios of 3:7 and 7:3, respectively. The objective is to find the mixing ratio of these alloys to achieve a new alloy with a gold to silver ratio of 2:3.

Mathematical Solution

Step 1: Defining the Alloys

Let's first define the fractions of gold and silver in the two alloys:

Alloy A: The gold to silver ratio is 3:7. Alloy B: The gold to silver ratio is 7:3.

The fractions of gold in each alloy are:

Alloy A: (frac{3}{10}) Alloy B: (frac{7}{10})

The fractions of silver in each alloy are complementing the gold fractions:

Alloy A: (frac{7}{10}) Alloy B: (frac{3}{10})

Step 2: Target Ratio

The target ratio for the final alloy is 2:3.

The fractions of gold and silver in the target alloy are:

Gold: (frac{2}{5}) Silver: (frac{3}{5})

Step 3: Applying Alligation

We use the alligation method to determine the mixing ratio:

Calculate the difference between the gold fraction in Alloy A and the target fraction: Calculate the difference between the gold fraction in Alloy B and the target fraction: The ratios of the difference amplify to the differences themselves.

Let's calculate the differences:

Alloy A: (frac{4}{10}) - (frac{3}{10}) (frac{1}{10}) Alloy B: (frac{7}{10}) - (frac{4}{10}) (frac{3}{10})

The ratio of Alloy A to Alloy B is the inverse of these differences:

Ratios: (frac{frac{3}{10}}{frac{1}{10}}) 3:1)

Hence, the two alloys must be mixed in the ratio 3:1 to achieve the desired 2:3 gold to silver ratio.

Conclusion

By understanding the alligation method and applying the principles of fraction conversion, we can determine the exact mixing ratio of two alloys to achieve a specific gold to silver ratio. In this case, the desired ratio of 2:3 can be obtained by mixing Alloy A and Alloy B in a 3:1 ratio.

Additional Insights

Intuitively, we can see that mixing three units of the 3:7 alloy with one unit of the 7:3 alloy will yield the desired 2:3 ratio. This is confirmed by the calculations shown above.

The alligation method is not only useful in metallurgy but can also be applied to other fields such as chemistry and finance where mixing different compositions to achieve a target mixture is necessary.

Conclusion

The two alloys must be mixed in the ratio of 3:1 to achieve a final alloy with a gold to silver ratio of 2:3. This solution provides a clear, methodical approach to solving similar problems in the realm of materials science and metallurgy.