A trader earns `20%` profit even after allowing `30%` discount to his customers. Minimum how much per cent does his balance weigh less than what it should weigh?

Options

A. 41.67
B. 55.55
C. 16.67
D. 56

Answer

To solve the problem step by step, we will analyze the information given and calculate the required percentage of weight loss.

### Step 1: Understand the Profit and Discount
The trader allows a discount of 30% on the selling price (SP) and still makes a profit of 20% on the cost price (CP).

### Step 2: Set Up the Equations
Let the cost price (CP) be 100 (for simplicity).
- Profit of 20% means the selling price (SP) is:
\[ SP = CP + 20\% \text{ of } CP = 100 + 20 = 120 \]

### Step 3: Calculate the Selling Price After Discount
The trader gives a discount of 30% on the selling price (SP):
- Discount = 30% of SP = 30% of 120 = 36
- Therefore, the selling price after discount (SP') is:
\[ SP' = SP - \text{Discount} = 120 - 36 = 84 \]

### Step 4: Relate Selling Price After Discount to Cost Price
Since the selling price after the discount is 84, we can set up the equation:
\[ SP' = CP + \text{Profit} \]
Where profit is still 20% of CP. Since CP is 100, the profit is 20. Thus:
\[ SP' = 100 + 20 = 120 \]
But we found SP' to be 84. This means the trader is effectively selling less weight than he should.

### Step 5: Calculate the Ratio of Selling Price to Cost Price
From the above, we can find the ratio of SP to CP:
\[ \frac{SP}{CP} = \frac{120}{100} = \frac{12}{10} = \frac{12}{7} \]

### Step 6: Calculate the Weight Loss
Let the actual weight that should be sold (based on CP) be represented as 12 units. The trader is selling only 7 units (due to the weight loss). The weight loss can be calculated as:
\[ \text{Weight Loss} = 12 - 7 = 5 \]

### Step 7: Calculate the Percentage of Weight Loss
To find the percentage of weight loss compared to the original weight:
\[ \text{Percentage Weight Loss} = \left(\frac{\text{Weight Loss}}{\text{Original Weight}}\right) \times 100 = \left(\frac{5}{12}\right) \times 100 \approx 41.67\% \]

### Conclusion
The minimum percentage that the balance weighs less than what it should weigh is approximately **41.67%**.

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