Understanding the Dealer's Article Sale

This problem involves a dealer selling articles in two distinct parts. A major portion (three-fourth or \(\frac{3}{4}\)) is sold at a profit, while the remaining portion is sold without any profit or loss (at the cost price). We need to determine the overall profit or gain percentage on the entire transaction.

Let's break down the transaction into parts to analyze the cost and selling price for each.

Calculating Cost Price and Selling Price

To make the calculation easier, we can assume a total number of articles and a cost price per article. However, working with fractions directly is more general.

• Let the total number of articles the dealer has be \(N\).
• Let the cost price (CP) of each article be \(C\).

So, the total cost price of all \(N\) articles is \( \text{Total CP} = N \times C \).

Part 1: Three-Fourth Articles

• Number of articles sold in this part: \(\frac{3}{4}\) of \(N\), which is \( \frac{3}{4}N \).
• Cost price of these articles: \( \frac{3}{4}N \times C \).
• These articles were sold at a gain of 20%. The selling price (SP) for this part is: \( \text{SP}_1 = \left(\frac{3}{4}N \times C\right) \times \left(1 + \frac{20}{100}\right) \) \( \text{SP}_1 = \left(\frac{3}{4}N \times C\right) \times (1 + 0.20) \) \( \text{SP}_1 = \left(\frac{3}{4}N \times C\right) \times 1.2 \) \( \text{SP}_1 = 0.9 N C \)

Part 2: Remaining Articles

• The remaining articles are \(1 - \frac{3}{4} = \frac{1}{4}\) of \(N\), which is \( \frac{1}{4}N \).
• Cost price of these articles: \( \frac{1}{4}N \times C \).
• These articles were sold at the cost price. So, the selling price (SP) for this part is: \( \text{SP}_2 = \frac{1}{4}N \times C \) \( \text{SP}_2 = 0.25 N C \)

Calculating Total Selling Price and Total Gain

The total selling price for the whole transaction is the sum of the selling prices from both parts:

\( \text{Total SP} = \text{SP}_1 + \text{SP}_2 \) \( \text{Total SP} = 0.9 N C + 0.25 N C \) \( \text{Total SP} = (0.9 + 0.25) N C \) \( \text{Total SP} = 1.15 N C \)

The total cost price was \( \text{Total CP} = N \times C \).

Since the Total SP (\(1.15 NC\)) is greater than the Total CP (\(NC\)), the dealer made a gain.

The total gain is: \( \text{Total Gain} = \text{Total SP} - \text{Total CP} \) \( \text{Total Gain} = 1.15 N C - N C \) \( \text{Total Gain} = (1.15 - 1) N C \) \( \text{Total Gain} = 0.15 N C \)

Calculating Overall Gain Percentage

The gain percentage is calculated based on the total cost price using the formula:

\( \text{Gain Percentage} = \left(\frac{\text{Total Gain}}{\text{Total CP}}\right) \times 100\% \)

Substitute the values we found: \( \text{Gain Percentage} = \left(\frac{0.15 N C}{N C}\right) \times 100\% \) \( \text{Gain Percentage} = 0.15 \times 100\% \) \( \text{Gain Percentage} = 15\% \)

Thus, the gain earned by the dealer in the whole transaction is 15%.

Revision Table: Key Concepts in Profit and Loss

Additional Information: Weighted Average Concept

This problem can also be viewed using the concept of a weighted average of profit percentages.

• Fraction of articles sold at 20% gain: \(\frac{3}{4}\)
• Fraction of articles sold at 0% gain (cost price): \(\frac{1}{4}\)

The overall gain percentage is the weighted average of the individual gain percentages:

Overall Gain % = \( \left(\frac{3}{4} \times 20\%\right) + \left(\frac{1}{4} \times 0\%\right) \)

Overall Gain % = \( \left(\frac{3}{4} \times 20\right)\% + 0\% \)

Overall Gain % = \( (3 \times 5)\% \)

Overall Gain % = \( 15\% \)

This weighted average method confirms the result obtained through detailed CP and SP calculation and is a quick way to solve such problems.