A trader earns `40%` profit by marking up his goods by `20%`. Minimum how much per cent does his balance weigh less than that it should weigh?

Options

A. 12
B. 14.28
C. 16.67
D. none of these

Answer

To solve the problem step by step, we will analyze the information given and apply the concepts of profit, loss, and markup.

### Step-by-Step Solution:

1. **Understanding Profit and Markup**:
- The trader earns a profit of 40%. This means if the cost price (CP) of an item is `x`, then the selling price (SP) can be calculated as:
\[ SP = CP + Profit = x + 0.4x = 1.4x \]

2. **Calculating the Marked Price**:
- The trader marks up his goods by 20%. If we assume the cost price is `x`, then the marked price (MP) is:
\[ MP = CP + Markup = x + 0.2x = 1.2x \]

3. **Setting Up the Relationship**:
- The selling price is also related to the marked price. Since the trader sells the goods at a selling price of `1.4x` (from step 1), we can set up the following equation:
\[ SP = MP \times (1 - Discount) \]
- However, since we are not given a discount, we can assume that the selling price is less than the marked price.

4. **Finding the Ratio of CP to SP**:
- From the profit percentage, we know:
\[ \frac{SP}{CP} = \frac{1.4x}{x} = 1.4 \]
- From the markup percentage, we have:
\[ \frac{MP}{CP} = \frac{1.2x}{x} = 1.2 \]

5. **Using the Ratio Method**:
- We can set up a ratio based on the weights. Let’s denote the actual weight of the goods that should be sold as `W` and the weight that is actually sold as `W'`.
- The profit indicates that for every `5` units of weight (CP), the trader is selling `7` units (SP). Thus:
\[ \frac{W'}{W} = \frac{5}{7} \]

6. **Calculating the Weight Difference**:
- If `W` is the expected weight (7 units), then the actual weight sold is:
\[ W' = \frac{5}{7}W \]

7. **Finding the Percentage Weight Loss**:
- The weight loss can be calculated as:
\[ \text{Weight Loss} = W - W' = W - \frac{5}{7}W = \frac{2}{7}W \]
- To find the percentage weight loss:
\[ \text{Percentage Weight Loss} = \left(\frac{\text{Weight Loss}}{W}\right) \times 100 = \left(\frac{\frac{2}{7}W}{W}\right) \times 100 = \frac{2}{7} \times 100 \approx 28.57\% \]

### Final Answer:
The minimum percentage that his balance weighs less than it should weigh is approximately **28.57%**.