In what proportion should Raj, Karan and Altaf distribute the profit among themselves? Statement I: Raj receives two fifths of the total profit. Statement II: Karan and Altaf together have contributed 75% of the total investment.

Introduction / Context:
This is a partnership style data sufficiency question. Raj, Karan and Altaf share some profit, and we are asked to find the proportion in which this profit is distributed among them. Two statements give partial information about Raj's share and about the combined investment shares of Karan and Altaf. The goal is to decide whether this information is sufficient to determine a unique profit sharing ratio Raj : Karan : Altaf.

Given Data / Assumptions:
- Question: In what proportion would Raj, Karan and Altaf distribute the profit?
- Statement I: Raj gets two fifths of the total profit.
- Statement II: Karan and Altaf together have made 75% of the total investment.
- Usual partnership assumption: if not stated otherwise, profit shares are typically proportional to investments, but here no direct explicit statement links the two in the given text.

Concept / Approach:
To determine the exact profit sharing ratio among three people, we need either direct information about each person's share of profit or a clear connection between profit and some other quantity (such as investment), along with the proportions of that quantity for each person. Data sufficiency analysis checks whether the statements provide enough information to obtain a unique triplet ratio Raj : Karan : Altaf. If there remain multiple consistent distributions, then the data are not sufficient.

Step-by-Step Solution:
Step 1: Use statement I alone. It tells us that Raj receives two fifths of the total profit, so Raj's profit share is 2/5, and the combined share of Karan and Altaf is 3/5. However, this does not tell us how that 3/5 is split between Karan and Altaf. Many different distributions such as Raj : Karan : Altaf = 2 : 1 : 2 or 2 : 2 : 1 would satisfy the condition that Raj has 2/5 of total profit. Thus, statement I alone is not sufficient. Step 2: Use statement II alone. It says that Karan and Altaf together contributed 75% of the total investment, so Raj contributed 25%. This allows us to express the investment proportions as 25% for Raj and 75% for Karan plus Altaf combined, but it does not state how that 75% is divided between Karan and Altaf, nor does it explicitly state that profit is distributed in proportion to investment. So statement II alone is not sufficient to determine the exact profit distribution. Step 3: Combine statements I and II. With both statements, we know Raj's investment share (25%) and Raj's profit share (2/5 = 40%). This suggests that Raj's profit share is not necessarily directly proportional to his investment share, and we still do not know how Karan and Altaf split either the investment or the profit between themselves. Step 4: Even if we assumed that profit and investment are related, we still lack any information that uniquely fixes the ratio between Karan and Altaf. Many combinations of Karan and Altaf shares can satisfy both the total investment constraint and the total profit constraint while changing the internal split between these two partners.

Verification / Alternative check:
Try constructing a concrete example. Suppose total profit is 1 unit. From statement I, Raj gets 2/5, so Karan plus Altaf together get 3/5. Now suppose total investment is also 1 unit. Statement II says Raj invests 1/4 and Karan plus Altaf together invest 3/4. You can choose infinitely many ways to split the 3/5 profit between Karan and Altaf while still keeping their combined investment at 3/4. This shows that the internal ratio Karan : Altaf is not determined uniquely, so the overall profit sharing ratio among all three remains undetermined.

Why Other Options Are Wrong:
- Option a is wrong because statement I alone does not fix how Karan and Altaf split the profit.
- Option b is wrong since statement II alone only gives combined investment share, not the profit distribution, and not the internal ratio between Karan and Altaf.
- Option c is incorrect because neither statement alone is sufficient; both leave multiple possibilities.
- Option d is wrong because even combining both statements still leaves us with more than one possible profit distribution ratio among the three partners.

Common Pitfalls:
A frequent mistake is to assume that knowing Raj's profit share and the combined investment share of Karan and Altaf automatically fixes all three profit shares. Another pitfall is to assume without careful thought that profit must always be proportional to investment, and that this somehow determines internal splits without additional data. In data sufficiency questions, you should avoid introducing extra assumptions and focus strictly on what can be logically inferred from the given statements.

Final Answer:
Even when both statements are considered together, the exact profit sharing ratio among Raj, Karan and Altaf cannot be determined uniquely. So the correct data sufficiency choice is option E.