A man buys some oranges at Rs. 5 per dozen and an equal number of oranges at Rs. 4 per dozen. He then sells all of the oranges at Rs. 5.50 per dozen and makes a profit of Rs. 50 in total. How many oranges does he buy?

Introduction / Context:
This problem combines weighted average cost and total profit to determine the number of items bought. The shopkeeper buys oranges at two different rates and sells all of them at a single selling price per dozen. The goal is to compute how many dozens, and hence how many oranges, he must have bought to achieve a stated profit.

Given Data / Assumptions:

• Some oranges are bought at Rs. 5 per dozen.
• An equal number of oranges are bought at Rs. 4 per dozen.
• All oranges are sold at Rs. 5.50 per dozen.
• Total profit earned = Rs. 50.
• We are asked how many oranges he buys, which is usually expressed in dozens first.

Concept / Approach:
Let the man buy n dozens at Rs. 5 per dozen and another n dozens at Rs. 4 per dozen. Then the total quantity is 2n dozens. We calculate total cost price, total selling price, and hence total profit in terms of n. Equating that profit to Rs. 50 allows us to solve for n. Finally we can express the answer either as dozens or individual oranges.

Step-by-Step Solution:
Step 1: Let number of dozens bought at Rs. 5 per dozen be n. Then the same number n dozens are bought at Rs. 4 per dozen.Step 2: Total cost price = 5n + 4n = 9n rupees.Step 3: Total quantity of oranges = 2n dozens.Step 4: Selling price per dozen = Rs. 5.50, so total selling price = 2n * 5.5 = 11n rupees.Step 5: Total profit = selling price - cost price = 11n - 9n = 2n rupees.Step 6: Given that total profit is Rs. 50, so 2n = 50 which implies n = 25 dozens at each rate.Step 7: Therefore total dozens bought = 2n = 50 dozens.

Verification / Alternative check:
If he buys 25 dozens at Rs. 5 and 25 dozens at Rs. 4, cost = 25 * 5 + 25 * 4 = 125 + 100 = Rs. 225.He sells 50 dozens at Rs. 5.50, so sales revenue = 50 * 5.5 = Rs. 275.Profit = 275 - 225 = Rs. 50, exactly as given in the question.

Why Other Options Are Wrong:
30, 40 or 60 dozens would produce total profits different from Rs. 50 when the same calculation is repeated. For example, 40 dozens total would give 2n = profit = 40 rupees, not Rs. 50. The option 20 dozens corresponds to n = 10, giving profit 20 rupees only. Only 50 dozens satisfies the profit condition.

Common Pitfalls:
Some learners treat the phrase “equal number” as equal cost instead of equal quantity. Others incorrectly divide the Rs. 50 profit by Rs. 5.50 directly without considering the mixed purchase prices. Introducing a variable n and writing separate expressions for cost and revenue is the most systematic and reliable approach.

Final Answer:
The man buys a total of 50 dozens of oranges.