A merchant has 40 kg sugar, part of which he sells at 12% gain and the rest at `28%`.He gains `18%` on the whole. Find the quantity sold at `12%` profit.

Answer

To solve the problem step by step, we will use the concept of mixtures and alligation.

### Step 1: Understand the given information
- Total sugar = 40 kg
- Part sold at 12% gain
- Part sold at 28% gain
- Overall gain = 18%

### Step 2: Set up the alligation formula
We will use the alligation method to find the ratio of the quantities sold at different profit percentages.

### Step 3: Calculate the differences
- Difference between the overall gain (18%) and the lower gain (12%):
\[ 18\% - 12\% = 6\% \]
- Difference between the higher gain (28%) and the overall gain (18%):
\[ 28\% - 18\% = 10\% \]

### Step 4: Set up the ratio
Using the differences calculated:
- The ratio of the quantities sold at 12% gain to the quantities sold at 28% gain is:
\[ \text{Ratio} = 10 : 6 \]
- Simplifying this ratio gives:
\[ 5 : 3 \]

### Step 5: Calculate the quantities based on the ratio
Let the quantity sold at 12% gain be \(5x\) and the quantity sold at 28% gain be \(3x\). Since the total quantity is 40 kg:
\[ 5x + 3x = 40 \]
\[ 8x = 40 \]
\[ x = 5 \]

### Step 6: Find the quantity sold at 12% profit
Now, substituting the value of \(x\) back into the expression for the quantity sold at 12% profit:
\[ \text{Quantity sold at 12% profit} = 5x = 5 \times 5 = 25 \text{ kg} \]

### Final Answer
The quantity sold at 12% profit is **25 kg**.

---