A marchant has 50 kg. of sugar, part of which he sells at 8% profit and the rest at 18% profit, He gains 14% on the whole. The quantity sold at 18% profit is:

Options

A. 20 kg
B. 30 kg
C. 15 kg
D. 35 kg

Answer

To solve the problem step by step, we can use the method of alligation. Here’s how to approach it:

### Step 1: Understand the given information
We have a total of 50 kg of sugar. The merchant sells part of it at an 8% profit and the rest at an 18% profit. The overall profit from selling all the sugar is 14%.

### Step 2: Set up the alligation
We will use alligation to find the ratio of the quantities sold at different profit percentages. The profits are:
- Profit 1 (P1) = 8%
- Profit 2 (P2) = 18%
- Overall Profit (P) = 14%

### Step 3: Calculate the differences
Now, we calculate the differences between the overall profit and the individual profits:
- Difference between P and P1: \( 14 - 8 = 6 \)
- Difference between P2 and P: \( 18 - 14 = 4 \)

### Step 4: Set up the ratio
Using the differences calculated, we can set up the ratio of the quantities sold at each profit:
- The ratio of the quantities sold at 8% profit to the quantities sold at 18% profit is given by the differences:
\[ \text{Ratio} = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Determine the total parts
The total parts in the ratio are \( 3 + 2 = 5 \).

### Step 6: Calculate the quantity corresponding to each part
Now, we know that the total quantity of sugar is 50 kg. We can find the value of one part:
\[ \text{Value of one part} = \frac{50 \text{ kg}}{5} = 10 \text{ kg} \]

### Step 7: Calculate the quantity sold at 18% profit
Since the ratio of the quantity sold at 8% profit to that sold at 18% profit is 3:2, the quantity sold at 18% profit corresponds to 2 parts:
\[ \text{Quantity sold at 18% profit} = 2 \times 10 \text{ kg} = 20 \text{ kg} \]

### Final Answer
The quantity sold at 18% profit is **20 kg**.

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