If Soham while selling two goats at the same price, makes a profit of 10% on one goat and suffers a loss of 10% on the other one (UPSC 2014)

To solve this problem, we'll break it down step by step:

1. Break Down the Problem

• Let the selling price (SP) of each goat be $S$.
• Let the cost price (CP) of the first goat be $CP_1$ and the cost price of the second goat be $CP_2$.
• A profit of 10% on the first goat means: $S = CP_1 + 0.1 \cdot CP_1 = 1.1 \cdot CP_1$
• A loss of 10% on the second goat means: $S = CP_2 - 0.1 \cdot CP_2 = 0.9 \cdot CP_2$

2. Relevant Concepts

From the above equations, we can relate the cost prices to the selling price:

1. From the first goat: $CP_1 = \frac{S}{1.1}$
2. From the second goat: $CP_2 = \frac{S}{0.9}$

3. Analysis and Detail

• The total cost price (CP) of both goats is: $CP_{total} = CP_1 + CP_2 = \frac{S}{1.1} + \frac{S}{0.9}$

• To combine these fractions, find a common denominator: $CP_{total} = S \left(\frac{0.9 + 1.1}{0.99}\right) = S \left(\frac{2}{0.99}\right) = \frac{2S}{0.99}$

• The total selling price (SP) of both goats is: $SP_{total} = S + S = 2S$

4. Verify and Summarize

• The overall profit or loss can be calculated as: $Profit\% = \frac{SP_{total} - CP_{total}}{CP_{total}} \times 100$ Substituting the values: $Profit\% = \frac{2S - \frac{2S}{0.99}}{\frac{2S}{0.99}} \times 100$ $= \frac{2S \left(0.99 - 1\right)}{\frac{2S}{0.99}} \times 100$ $= \frac{2S \cdot (-0.01)}{\frac{2S}{0.99}} \times 100$ $= -0.01 \cdot 0.99 \times 100 = -0.99\%$

Final Answer

Soham incurs an overall loss of approximately 0.99%.