Two metals X and Y are to be used for making two different alloys. If the ratio by weight of X : Y in the first alloy is 6 : 5 and that in the second is 7 : 13, how many kg of X metal must be melted along with 11 kg of the first alloy and 20 kg of the second so as to produce a new alloy containing `40%` of metal Y?

Options

A. 11
B. 12
C. 13
D. 14

Answer

To solve the problem, we need to find out how many kg of metal X must be melted along with 11 kg of the first alloy and 20 kg of the second alloy to produce a new alloy containing 40% of metal Y.

### Step-by-Step Solution:

1. **Determine the Composition of the First Alloy:**
- The ratio of X to Y in the first alloy is 6:5.
- Total parts = 6 + 5 = 11 parts.
- Weight of the first alloy = 11 kg.
- Weight of X in the first alloy = (6/11) * 11 kg = 6 kg.
- Weight of Y in the first alloy = (5/11) * 11 kg = 5 kg.

2. **Determine the Composition of the Second Alloy:**
- The ratio of X to Y in the second alloy is 7:13.
- Total parts = 7 + 13 = 20 parts.
- Weight of the second alloy = 20 kg.
- Weight of X in the second alloy = (7/20) * 20 kg = 7 kg.
- Weight of Y in the second alloy = (13/20) * 20 kg = 13 kg.

3. **Calculate the Total Amount of X and Y in Both Alloys:**
- Total weight of X from both alloys = 6 kg (from first alloy) + 7 kg (from second alloy) = 13 kg.
- Total weight of Y from both alloys = 5 kg (from first alloy) + 13 kg (from second alloy) = 18 kg.

4. **Determine the Total Weight of the New Alloy:**
- Total weight of the new alloy = weight of the first alloy + weight of the second alloy + weight of X to be melted.
- Let the weight of X to be melted be denoted as \( x \).
- Total weight of the new alloy = 11 kg + 20 kg + \( x \) kg = \( 31 + x \) kg.

5. **Set Up the Equation for the New Alloy:**
- According to the problem, the new alloy must contain 40% of metal Y.
- Therefore, the weight of Y in the new alloy = 40% of total weight of the new alloy.
- This can be expressed as:
\[ 0.4 \times (31 + x) = 18 \text{ kg (weight of Y from both alloys)} \]

6. **Solve for \( x \):**
\[ 0.4(31 + x) = 18 \]
\[ 31 + x = \frac{18}{0.4} \]
\[ 31 + x = 45 \]
\[ x = 45 - 31 \]
\[ x = 14 \text{ kg} \]

### Final Answer:
The amount of metal X that must be melted is **14 kg**.

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