Two alloys contain iron and lead in the ratio of `4 : 3 and 1 : 6` respectively. 7 kg of the first alloy, 14 kg of the second alloy and some amount of pure iron are mixed to from a new alloy. The resultant alloy has iron and lead in the ratio of `1 : 1`. Find the weight of the resultant alloy.

Answer

To solve the problem step by step, we need to calculate the amount of iron and lead in each alloy and then determine the total weight of the resultant alloy.

### Step 1: Calculate the amount of iron and lead in the first alloy.
The first alloy has iron and lead in the ratio of 4:3.

Given:
- Total weight of the first alloy = 7 kg
- Ratio of iron to lead = 4:3

To find the weight of iron in the first alloy:
- Total parts in the ratio = 4 + 3 = 7 parts
- Weight of iron = (4/7) * 7 kg = 4 kg
- Weight of lead = (3/7) * 7 kg = 3 kg

### Step 2: Calculate the amount of iron and lead in the second alloy.
The second alloy has iron and lead in the ratio of 1:6.

Given:
- Total weight of the second alloy = 14 kg
- Ratio of iron to lead = 1:6

To find the weight of iron in the second alloy:
- Total parts in the ratio = 1 + 6 = 7 parts
- Weight of iron = (1/7) * 14 kg = 2 kg
- Weight of lead = (6/7) * 14 kg = 12 kg

### Step 3: Calculate the total amount of iron and lead in the mixture.
Now, we will add the amounts of iron and lead from both alloys.

- Total weight of iron from both alloys = 4 kg (from first alloy) + 2 kg (from second alloy) = 6 kg
- Total weight of lead from both alloys = 3 kg (from first alloy) + 12 kg (from second alloy) = 15 kg

### Step 4: Set up the equation for the resultant alloy.
Let the amount of pure iron added be \( x \) kg. The resultant alloy has iron and lead in the ratio of 1:1.

According to the problem:
- Total weight of iron in the resultant alloy = 6 kg + \( x \)
- Total weight of lead in the resultant alloy = 15 kg

Since the ratio of iron to lead in the resultant alloy is 1:1, we can set up the equation:
\[ 6 + x = 15 \]

### Step 5: Solve for \( x \).
Now, we solve the equation:
\[ x = 15 - 6 \]
\[ x = 9 \]

### Step 6: Calculate the total weight of the resultant alloy.
Now, we can find the total weight of the resultant alloy:
- Total weight of the resultant alloy = weight of iron + weight of lead
- Total weight = (6 kg + 9 kg) + 15 kg = 30 kg

Thus, the weight of the resultant alloy is **30 kg**.