Understanding the Problem: Gain and Loss Calculation

The problem involves a man selling two articles at the same price, Rs. 9,975 each. On one article, he makes a profit (gain) of 5%, and on the other, he incurs a loss of 5%. We need to find the overall gain or loss on the entire transaction.

To find the overall gain or loss, we must first determine the cost price (CP) of each article. We are given the selling price (SP) and the percentage gain or loss for each.

Calculating Cost Price (CP) for Each Article

The formula relating Selling Price (SP), Cost Price (CP), and Percentage Gain/Loss is:

• If there is a gain: $\text{SP} = \text{CP} \times (1 + \frac{\text{Gain}\%}{100})$
• If there is a loss: $\text{SP} = \text{CP} \times (1 - \frac{\text{Loss}\%}{100})$

Article 1: Sold at 5% Gain

Selling Price (SP1) = Rs. 9,975

Gain% = 5%

Let the Cost Price of Article 1 be CP1.

Using the gain formula:

$\text{SP1} = \text{CP1} \times (1 + \frac{5}{100})$

$9975 = \text{CP1} \times (1 + 0.05)$

$9975 = \text{CP1} \times 1.05$

To find CP1, we rearrange the formula:

$\text{CP1} = \frac{9975}{1.05}$

$\text{CP1} = 9500$

So, the cost price of the first article is Rs. 9,500.

Article 2: Sold at 5% Loss

Selling Price (SP2) = Rs. 9,975

Loss% = 5%

Let the Cost Price of Article 2 be CP2.

Using the loss formula:

$\text{SP2} = \text{CP2} \times (1 - \frac{5}{100})$

$9975 = \text{CP2} \times (1 - 0.05)$

$9975 = \text{CP2} \times 0.95$

To find CP2, we rearrange the formula:

$\text{CP2} = \frac{9975}{0.95}$

$\text{CP2} = 10500$

So, the cost price of the second article is Rs. 10,500.

Calculating Overall Gain or Loss

Now, let's calculate the total selling price and the total cost price for both articles.

Total Selling Price (Total SP) = SP1 + SP2

Total SP = $9975 + 9975 = 19950$

Total Cost Price (Total CP) = CP1 + CP2

Total CP = $9500 + 10500 = 20000$

To find the overall gain or loss, we compare the Total SP and Total CP.

• If Total SP > Total CP, there is an overall gain.
• If Total SP < Total CP, there is an overall loss.
• If Total SP = Total CP, there is neither gain nor loss.

In this case, Total SP (19950) < Total CP (20000). This means there is an overall loss.

Overall Loss = Total CP - Total SP

Overall Loss = $20000 - 19950 = 50$

The overall loss is Rs. 50.

Summary of Calculations

Total Selling Price = Rs. 19,950

Total Cost Price = Rs. 20,000

Since Total CP > Total SP, there is a loss.

Overall Loss = Total CP - Total SP = $20000 - 19950 = \text{Rs. } 50$

Therefore, the man incurs an overall loss of Rs. 50 on the entire transaction of selling the two articles.

Revision Table: Profit and Loss Key Concepts

These formulas are essential for solving profit and loss problems like finding the overall gain or loss.

Additional Information: Special Case in Gain and Loss

This problem is an example of a special case in profit and loss calculations. When two articles are sold at the same selling price, and the gain percentage on one article is equal to the loss percentage on the other article (say, R%), there is always an overall loss.

The percentage loss in such a scenario can be calculated directly using the formula:

Overall Loss Percentage $= (\frac{R}{10})^2 \%$

In this specific problem, R = 5%.

Overall Loss Percentage $= (\frac{5}{10})^2 \% = (\frac{1}{2})^2 \% = \frac{1}{4} \% = 0.25 \%$

To find the amount of loss, we need the total cost price or total selling price. We found the Total CP = Rs. 20,000.

Overall Loss Amount = Overall Loss Percentage of Total CP

Overall Loss Amount $= 0.25 \%$ of $20000$

Overall Loss Amount $= \frac{0.25}{100} \times 20000 = \frac{1}{400} \times 20000 = \frac{20000}{400} = 50$

This confirms our calculated overall loss of Rs. 50, demonstrating how this special case formula can be used as a shortcut or a check.