A shopkeeper sells 100 kg of sugar partly at 10% profit and the remaining at 20% profit. If he gains 12% on the whole transaction, how much sugar did he sell at 20% profit?

Options

A. 25 kg
B. 40 kg
C. 20 kg
D. 30 kg

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = the amount of sugar sold at 20% profit (in kg)
- \( 100 - x \) = the amount of sugar sold at 10% profit (in kg)

### Step 2: Set Up the Cost Price
Assume the cost price of sugar per kg is \( Y \). Therefore, the total cost price of 100 kg of sugar is:
\[ \text{Total Cost Price} = 100Y \]

### Step 3: Calculate Selling Prices
- Selling price for \( x \) kg of sugar sold at 20% profit:
\[ \text{Selling Price at 20%} = x \times Y \times (1 + 0.20) = x \times Y \times 1.20 = 1.2xY \]

- Selling price for \( 100 - x \) kg of sugar sold at 10% profit:
\[ \text{Selling Price at 10%} = (100 - x) \times Y \times (1 + 0.10) = (100 - x) \times Y \times 1.10 = 1.1(100 - x)Y \]

### Step 4: Total Selling Price
The total selling price (SP) of the sugar is the sum of the selling prices:
\[ \text{Total Selling Price} = 1.2xY + 1.1(100 - x)Y \]

### Step 5: Gain on the Whole Transaction
The shopkeeper gains 12% on the whole transaction, so:
\[ \text{Total Selling Price} = \text{Total Cost Price} + 12\% \text{ of Total Cost Price} \]
\[ \text{Total Selling Price} = 100Y + 0.12 \times 100Y = 100Y \times 1.12 = 112Y \]

### Step 6: Set Up the Equation
Now, we can set the total selling price equal to the expression we derived:
\[ 1.2xY + 1.1(100 - x)Y = 112Y \]

### Step 7: Simplify the Equation
Dividing the entire equation by \( Y \) (assuming \( Y \neq 0 \)):
\[ 1.2x + 1.1(100 - x) = 112 \]
Expanding this gives:
\[ 1.2x + 110 - 1.1x = 112 \]

### Step 8: Combine Like Terms
Combine the \( x \) terms:
\[ (1.2 - 1.1)x + 110 = 112 \]
\[ 0.1x + 110 = 112 \]

### Step 9: Solve for \( x \)
Subtract 110 from both sides:
\[ 0.1x = 2 \]
Now divide by 0.1:
\[ x = \frac{2}{0.1} = 20 \]

### Conclusion
The amount of sugar sold at 20% profit is \( \boxed{20 \text{ kg}} \).

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