The ratio of copper and nickel by weight in the two alloys X and Y are 2:7 and 5:4. How many kilogram of the alloy X and Y are required to make 42 kg of new alloy Z in Which the ratio of copper and nickel is same?

Options

A. 6 kg and 36 kg
B. 10 kg and 32 kg
C. 7 kg and 35 kg
D. None of these

Answer

To solve the problem, we need to find how many kilograms of alloys X and Y are required to create 42 kg of a new alloy Z, where the ratio of copper to nickel is the same.

### Step-by-Step Solution:

1. **Identify the Ratios of Alloys X and Y**:
- The ratio of copper to nickel in alloy X is 2:7.
- The ratio of copper to nickel in alloy Y is 5:4.

2. **Calculate the Fraction of Copper in Each Alloy**:
- For alloy X:
- Total parts = 2 + 7 = 9
- Fraction of copper in X = \( \frac{2}{9} \)

- For alloy Y:
- Total parts = 5 + 4 = 9
- Fraction of copper in Y = \( \frac{5}{9} \)

3. **Determine the Desired Ratio for Alloy Z**:
- The problem states that the ratio of copper to nickel in alloy Z is 1:1.
- This means that the fraction of copper in Z is \( \frac{1}{2} \) (since 1 part copper and 1 part nickel makes 2 parts total).

4. **Set Up the Allegation Method**:
- We will use the allegation method to find the ratio of the quantities of alloys X and Y needed to achieve the desired ratio in alloy Z.

- The fractions are:
- Copper in X: \( \frac{2}{9} \)
- Copper in Y: \( \frac{5}{9} \)
- Copper in Z: \( \frac{1}{2} \)

5. **Calculate the Differences**:
- Difference between copper in Y and Z:
\[ \frac{5}{9} - \frac{1}{2} = \frac{10}{18} - \frac{9}{18} = \frac{1}{18} \]

- Difference between copper in Z and X:
\[ \frac{1}{2} - \frac{2}{9} = \frac{9}{18} - \frac{4}{18} = \frac{5}{18} \]

6. **Determine the Ratio of Alloys X and Y**:
- The ratio of the weights of X to Y is given by the inverse of the differences calculated:
\[ \text{Ratio of X to Y} = \frac{1/18}{5/18} = \frac{1}{5} \]
- This means for every 1 kg of X, we need 5 kg of Y.

7. **Calculate the Total Weight of Alloys**:
- Let the weight of alloy X be \( x \) kg and the weight of alloy Y be \( y \) kg.
- From the ratio, we have \( y = 5x \).
- The total weight of the new alloy Z is given as 42 kg:
\[ x + y = 42 \]
- Substituting \( y \) in the equation:
\[ x + 5x = 42 \implies 6x = 42 \implies x = 7 \]
- Therefore, \( y = 5 \times 7 = 35 \).

8. **Final Weights of Alloys X and Y**:
- The required weights are:
- Weight of alloy X = 7 kg
- Weight of alloy Y = 35 kg

### Conclusion:
The required amounts of alloys X and Y to make 42 kg of alloy Z are:
- Alloy X: 7 kg
- Alloy Y: 35 kg