In an alloy, lead and tin are in the ratio of 2 : 3. In the second alloy, the ratio of same elements is 3 : 4. If equal quantities of these two alloy are mixed to form a new alloy, then what will be the ratio of these two elements in the new alloy?

Options

A. `1:3`
B. `29:41`
C. `25:37`
D. `31:43`

Answer

To solve the problem step by step, we will analyze the ratios of lead and tin in both alloys and then combine them to find the new ratio in the resulting alloy.

### Step 1: Understand the Ratios in Each Alloy
- In the first alloy, lead and tin are in the ratio of 2:3. This means for every 2 parts of lead, there are 3 parts of tin.
- In the second alloy, lead and tin are in the ratio of 3:4. This means for every 3 parts of lead, there are 4 parts of tin.

### Step 2: Find the Total Parts in Each Alloy
- For the first alloy, the total parts = 2 (lead) + 3 (tin) = 5 parts.
- For the second alloy, the total parts = 3 (lead) + 4 (tin) = 7 parts.

### Step 3: Equalize the Quantities of Both Alloys
Since we are mixing equal quantities of both alloys, we need to express the quantities of lead and tin in a common unit. We can do this by finding a common multiple of the total parts.

- The least common multiple (LCM) of 5 and 7 is 35.
- To express the first alloy in terms of 35 parts:
- Lead in the first alloy = \( \frac{2}{5} \times 35 = 14 \)
- Tin in the first alloy = \( \frac{3}{5} \times 35 = 21 \)

- To express the second alloy in terms of 35 parts:
- Lead in the second alloy = \( \frac{3}{7} \times 35 = 15 \)
- Tin in the second alloy = \( \frac{4}{7} \times 35 = 20 \)

### Step 4: Combine the Quantities from Both Alloys
Now, we can combine the quantities of lead and tin from both alloys:
- Total lead = Lead from first alloy + Lead from second alloy = 14 + 15 = 29
- Total tin = Tin from first alloy + Tin from second alloy = 21 + 20 = 41

### Step 5: Write the Final Ratio
The ratio of lead to tin in the new alloy is:
- Ratio = Total lead : Total tin = 29 : 41

Thus, the final answer is that the ratio of lead to tin in the new alloy is **29:41**.

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