In a joint business, P and Q share profit according to their capital time contributions. What is the difference between the profit shares of P and Q at the end of one year? (a) P invests Rs. 80,000 and withdraws Rs. 20,000 after 6 months. (b) Q joins 4 months after the start of the business. (c) During the last 6 months, Q's capital is 80 percent of P's capital. Based on this information, can the difference in their profit shares be determined?

Introduction / Context:
This is a data sufficiency question involving capital time in a partnership. Instead of asking for the actual numerical difference between the profits of P and Q, the question focuses on whether the given statements provide enough information to determine that difference. Understanding what information is essential and what is missing is the key to solving such questions.

Given Data / Assumptions:

• We want the difference between the profit shares of P and Q at the end of one year.
• Statement (a): P invests Rs. 80,000 and withdraws Rs. 20,000 after 6 months (so his last six months capital is Rs. 60,000).
• Statement (b): Q joins the business 4 months after the start.
• Statement (c): During the last 6 months, Q's capital is 80 percent of P's capital.
• Profit is proportional to capital time.

Concept / Approach:
To determine the exact difference in profit between P and Q in rupees, we must know both the ratio of their capital time contributions and the total profit amount. The given statements provide some information about their capital time contributions but not the total profit. We will analyze each statement and combination to see if they allow computation of the monetary difference between their shares.

Step-by-Step Solution:
Step 1: From statement (a), P's capital is Rs. 80,000 for the first 6 months and Rs. 60,000 for the last 6 months.Step 2: This allows us to compute P's capital time but provides no information about Q's capital or the total profit.Step 3: From statement (b), we only know that Q joins after 4 months, so Q is in the business for 8 months.Step 4: Statement (b) alone does not give Q's capital amount or any relation to P's capital.Step 5: From statement (c), we know that in the last 6 months, Q's capital is 80 percent of P's capital, meaning Q's capital is 0.8 times P's capital for that period.Step 6: Combining (a), (b) and (c), we can determine the ratio of the capital time contributions of P and Q, and thus the ratio of their profit shares.Step 7: However, we still do not know the total profit of the business, so while the ratio of their shares can be computed, the absolute difference in rupees cannot be determined.Step 8: Therefore, the data is insufficient to find a unique numerical value for the difference between their profit shares.

Verification / Alternative check:
Even if we assume a total profit of T rupees and compute P : Q ratio from the given information, the difference between their shares will be some fraction of T. Without a specific value for T, the difference in rupees remains unknown and could vary with any change in total profit. This confirms that the given statements, even taken together, do not uniquely determine the required difference.

Why Other Options Are Wrong:
'Only a' and 'Only c' are incorrect because neither provides enough information about Q's investment duration and the total profit. 'Both a and b' is insufficient because we still lack the exact value of Q's capital or total profit. 'None of these' is not correct since one of the listed options correctly describes the situation: the data is not sufficient.

Common Pitfalls:
Many learners confuse knowing the ratio of profit shares with knowing the numerical difference in rupees. Another frequent mistake is to assume some arbitrary total profit to compute a difference, which violates the requirement of uniqueness in data sufficiency questions. Always check whether the total profit or an equivalent piece of information is available before concluding that the difference can be determined.

Final Answer:
The correct conclusion is that the data is not sufficient to determine the numeric difference in profit shares between P and Q.