In an alloy of German silver, the ratio of copper to zinc was 21 : 16 while that of nickel to zinc was 7:24. What was the ratio of copper to zinc to nickel in the alloy?

Options

A. `21:16:7`
B. `17:21:4`
C. `63:48:14`
D. `68:28:21`

Answer

To find the ratio of copper to zinc to nickel in the alloy, we will follow these steps:

1. **Identify the given ratios**:
- The ratio of copper (C) to zinc (Z) is given as 21:16.
- The ratio of nickel (N) to zinc (Z) is given as 7:24.

2. **Express the ratios in terms of a common variable**:
- Let the amount of zinc (Z) be represented as a common variable.
- From the copper to zinc ratio (C:Z = 21:16), we can express:
- C = 21k (where k is a common multiplier)
- Z = 16k

- From the nickel to zinc ratio (N:Z = 7:24), we can express:
- N = 7m (where m is another common multiplier)
- Z = 24m

3. **Equate the two expressions for zinc (Z)**:
- Since both expressions represent the amount of zinc, we can set them equal to each other:
- 16k = 24m

4. **Solve for k in terms of m**:
- Rearranging the equation gives:
- k/m = 24/16
- k/m = 3/2
- k = (3/2)m

5. **Substitute k back into the expressions for C and Z**:
- Substitute k into the expressions for C and Z:
- C = 21k = 21(3/2)m = 31.5m
- Z = 16k = 16(3/2)m = 24m
- N = 7m

6. **Write the final ratio of C:Z:N**:
- Now we have:
- C = 31.5m
- Z = 24m
- N = 7m
- The ratio of C:Z:N can be written as:
- C:Z:N = 31.5m : 24m : 7m
- This simplifies to:
- C:Z:N = 31.5 : 24 : 7

7. **Convert to whole numbers**:
- To eliminate the decimal, multiply each term by 2:
- C:Z:N = 63 : 48 : 14

8. **Final ratio**:
- The final ratio of copper to zinc to nickel in the alloy is:
- C:Z:N = 63 : 48 : 14