Understanding the Ratios in a German Silver Alloy

This problem involves finding the combined ratio of three components – copper, zinc, and nickel – in an alloy known as German Silver. We are given the ratios of two pairs of these components, each involving zinc.

Given Ratios

We are provided with the following information:

• Ratio of Copper to Zinc (Cu : Zn) is 17 : 7
• Ratio of Nickel to Zinc (Ni : Zn) is 4 : 3

Our goal is to find the ratio of Copper to Zinc to Nickel (Cu : Zn : Ni).

Finding the Combined Ratio (Copper : Zinc : Nickel)

To combine these two ratios into a single ratio involving all three components, we need to make the value corresponding to the common element, which is Zinc (Zn), the same in both ratios.

The zinc values in the given ratios are 7 (in Cu : Zn) and 3 (in Ni : Zn). We need to find the least common multiple (LCM) of 7 and 3 to make the zinc value consistent.

The LCM of 7 and 3 is \(7 \times 3 = 21\).

Adjusting the Ratios

Now, we adjust each given ratio so that the part corresponding to Zinc becomes 21.

1. For the ratio Cu : Zn = 17 : 7:
   To make the zinc part 21, we need to multiply 7 by \( \frac{21}{7} = 3 \). We must multiply both parts of the ratio by 3 to maintain the proportion.
   New Cu : Zn ratio = \( (17 \times 3) : (7 \times 3) \) = 51 : 21.
2. For the ratio Ni : Zn = 4 : 3:
   To make the zinc part 21, we need to multiply 3 by \( \frac{21}{3} = 7 \). We must multiply both parts of the ratio by 7 to maintain the proportion.
   New Ni : Zn ratio = \( (4 \times 7) : (3 \times 7) \) = 28 : 21.

Combining the Adjusted Ratios

Now we have the ratios with a consistent value for Zinc:

• Cu : Zn = 51 : 21
• Ni : Zn = 28 : 21

We can rewrite the second ratio as Zn : Ni = 21 : 28.

Since the value for Zinc is the same (21) in both adjusted ratios, we can combine them directly to find the ratio of Copper to Zinc to Nickel (Cu : Zn : Ni).

Cu : Zn : Ni = 51 : 21 : 28.

Comparing with Options

Let's compare our calculated ratio with the given options:

Our calculated ratio, 51 : 21 : 28, matches Option 2.

Step-by-Step Summary

Here’s a quick summary of the steps taken:

1. Identify the common element (Zinc) in the given ratios.
2. Find the LCM of the values of the common element (LCM of 7 and 3 is 21).
3. Adjust the first ratio (Cu : Zn = 17 : 7) by multiplying both parts by \( \frac{21}{7} = 3 \) to get 51 : 21.
4. Adjust the second ratio (Ni : Zn = 4 : 3) by multiplying both parts by \( \frac{21}{3} = 7 \) to get 28 : 21.
5. Combine the adjusted ratios using the common zinc value (21) to get Cu : Zn : Ni = 51 : 21 : 28.

Revision Table: German Silver Ratio Problem

Additional Information: German Silver Alloy and Ratios

What is German Silver?

German Silver is actually an alloy, not pure silver. It typically consists of copper, zinc, and nickel. Despite its name, it contains no silver. The proportion of these metals can vary, leading to different properties of the alloy, such as hardness, corrosion resistance, and color.

Importance of Ratios in Alloys

Ratios are crucial in chemistry and metallurgy for defining the composition of mixtures and alloys. The specific ratio of components in an alloy determines its physical and chemical properties. For example, changing the ratio of copper, zinc, and nickel in German Silver can affect its color, workability, and melting point.

Combining Ratios Technique

The method used in this problem, finding a common value for the connecting element (Zinc in this case), is a standard technique for combining ratios. This technique is widely applicable in problems involving proportions of three or more quantities where pairwise ratios are given.