A farmer buys one goat and one sheep together for Rs. 3500. He sells the sheep at a profit of 20% and the goat at a loss of 10%. If both animals are sold at exactly the same selling price, what is the cost price in rupees of the cheaper animal?

Introduction / Context:
This profit and loss question involves two animals bought together for a single combined cost price and then sold with different gain and loss percentages. The twist is that both animals are sold at exactly the same selling price, which creates a useful algebraic relationship between their individual cost prices. Problems like this are common in competitive exams because they test comfort with percentage profit and loss, as well as forming and solving simple linear equations. Understanding how to express profit and loss in terms of cost price and selling price is the key to unlocking such questions quickly and accurately.

Given Data / Assumptions:
- The farmer buys one goat and one sheep for a total cost price of Rs. 3500. - The sheep is sold at a profit of 20% on its cost price. - The goat is sold at a loss of 10% on its cost price. - Both animals are sold at the same selling price. - We are asked to find the cost price of the cheaper animal.

Concept / Approach:
To solve this problem, we let the unknown cost prices of the goat and the sheep be variables and then use two equations: one from the total cost and another from the equality of selling prices. Profit and loss percentages are always applied on cost price. So, selling price = cost price * (1 + profit%) or cost price * (1 - loss%). Because the farmer sells both animals at the same price, those two expressions for selling price must be equal. Solving these equations together will give the individual cost prices, and from them we can identify the cheaper animal. This approach avoids guesswork and ensures a precise answer.

Step-by-Step Solution:
Let the cost price of the goat be G rupees. Let the cost price of the sheep be S rupees. From the total cost: G + S = 3500. Sheep is sold at 20% profit, so its selling price = S * 1.20 = 1.2S. Goat is sold at 10% loss, so its selling price = G * 0.90 = 0.9G. Given that the selling prices are equal, we have 1.2S = 0.9G. Rearrange: G = (1.2 / 0.9) * S = (4 / 3) * S. Substitute into G + S = 3500: (4/3)S + S = 3500. Combine: (4/3)S + (3/3)S = (7/3)S = 3500. So S = 3500 * (3/7) = 1500. Then G = 3500 - 1500 = 2000. The two cost prices are Rs. 1500 (sheep) and Rs. 2000 (goat); the cheaper animal costs Rs. 1500.

Verification / Alternative check:
Check selling price of sheep: 20% profit on Rs. 1500 means profit = 0.20 * 1500 = Rs. 300, so selling price = 1500 + 300 = Rs. 1800. Check selling price of goat: 10% loss on Rs. 2000 means loss = 0.10 * 2000 = Rs. 200, so selling price = 2000 - 200 = Rs. 1800. Both selling prices are indeed equal (Rs. 1800 each), so the calculations are consistent with the condition given in the question.

Why Other Options Are Wrong:
- Rs. 2000: This is the cost of the more expensive animal, not the cheaper one. - Rs. 1750 and Rs. 2250: These values do not satisfy both equations G + S = 3500 and equal selling prices with 20% profit and 10% loss respectively. - Rs. 2500: This would force the other animal's cost price to be Rs. 1000, which fails the equal-selling-price condition under the given profit and loss percentages.

Common Pitfalls:
Many learners forget that profit and loss percentages are always calculated on cost price, not on selling price. Another common mistake is to assume the cheaper animal is the one sold at loss without actually solving the equations. Some students simply average the profit and loss percentages, which is incorrect because the items have different cost prices. It is also easy to forget to use the total cost equation G + S = 3500 alongside the equal selling price equation, but both are necessary to get unique values.

Final Answer:
The cost price of the cheaper animal is Rs. 1500.