Correct Option (3)

When two items are sold at the same price, and one yields a profit of 10% while the other incurs a loss of 10%, there is always an overall loss. This can be demonstrated through the following calculation:

Let the Cost Price (CP) of the first goat be ₹ x.
Selling Price (SP) of the first goat (10% profit) = $x + \frac{10}{100} x = \frac{110}{100} x$ .
Let the Cost Price (CP) of the second goat be ₹ y.
Selling Price (SP) of the second goat (10% loss) = $y - \frac{10}{100} y = \frac{90}{100} y$ .

Given that the selling prices are equal:

\frac{110}{100} x = \frac{90}{100} y

x = \frac{90}{110} y = \frac{9}{11} y

Now, calculate the total Selling Price (SP) and total Cost Price (CP):

Total SP = SP of first goat + SP of second goat = $\frac{110}{100} x + \frac{90}{100} y$ .
Substitute $x = \frac{9}{11} y$ : Total SP = $\frac{110}{100} \times \frac{9}{11} y + \frac{90}{100} y = \frac{9}{10} y + \frac{90}{100} y = \frac{90}{100} y + \frac{90}{100} y = \frac{180}{100} y = \frac{18}{10} y$ .
Total CP = CP of first goat + CP of second goat = $x + y$ .
Substitute $x = \frac{9}{11} y$ : Total CP = $\frac{9}{11} y + y = \frac{9 y + 11 y}{11} = \frac{20}{11} y$ .

To determine profit or loss, compare Total CP and Total SP:

Total CP = $\frac{20}{11}y \approx 1.818y$
Total SP = $\frac{18}{10}y = 1.8y$

Since Total CP > Total SP, there is an overall loss.

Calculate the loss amount:

Loss = Total CP - Total SP = \frac{20}{11} y - \frac{18}{10} y = \frac{200 y - 198 y}{110} = \frac{2}{110} y

Calculate the percentage loss:

Loss % = (\frac{Loss}{Total CP}) \times 100

Loss % = (\frac{\frac{2}{110} y}{\frac{20}{11} y}) \times 100 = (\frac{2}{110} \times \frac{11}{20}) \times 100

\[\text{Loss \%} = \left( \frac{1}{10} \times \frac{1}{10} \right) \times 100 = \frac{1}{100} \times 100 = 1\%\]

Therefore, Sohan suffers a loss of 1%.

Incorrect Options:

Option 1 (he makes no profit and no loss): This is incorrect. In scenarios where two articles are sold at the same price, and one is sold at a profit of P% and the other at a loss of P%, there is always an overall loss. The condition of equal selling prices prevents a no-profit, no-loss outcome.
Option 2 (he makes a profit of 1%): This is incorrect. As demonstrated in the detailed calculation for the correct option, the total cost price of the two goats exceeds their total selling price. This indicates an overall loss, not a profit.
Option 4 (he suffers a loss of 2%): This is incorrect. The precise calculation shows that the overall loss percentage is 1%, not 2%. The discrepancy arises from the specific relationship between the cost prices and selling prices when the selling prices are equal.