A and B invest Rs. 10,000 each in a business. A keeps his capital for 8 months, while B keeps his capital invested for the full 12 months. If the total profit at the end of the year is Rs. 25,000, what are the respective shares of A and B?

Introduction / Context:
This is a straightforward partnership problem where two partners invest the same amount but for different periods of time. The task is to find their individual shares in the final profit by using the idea of capital time products and proportional distribution.

Given Data / Assumptions:

• A invests Rs. 10,000 for 8 months.
• B invests Rs. 10,000 for 12 months.
• Total profit at the end of the year is Rs. 25,000.
• We must find the profit shares of A and B.

Concept / Approach:
In partnership questions, profit is divided in proportion to the product of capital and time for each partner. Since both partners have equal capital but for different durations, the ratio of their shares will simply match the ratio of their investment periods. Once we know the ratio, we split the total profit according to that ratio.

Step-by-Step Solution:
Step 1: Calculate A's capital time product: 10,000 * 8 = 80,000. Step 2: Calculate B's capital time product: 10,000 * 12 = 1,20,000. Step 3: Ratio of A : B = 80,000 : 1,20,000. Step 4: Simplify this ratio by dividing both sides by 40,000 to get 2 : 3. Step 5: Total number of ratio parts = 2 + 3 = 5. Step 6: Total profit is Rs. 25,000, so each part is 25,000 / 5 = Rs. 5000. Step 7: A's share = 2 parts = 2 * 5000 = Rs. 10,000. Step 8: B's share = 3 parts = 3 * 5000 = Rs. 15,000.

Verification / Alternative check:
If A receives Rs. 10,000 and B receives Rs. 15,000, their total is 10,000 + 15,000 = Rs. 25,000, which equals the given total profit. The profit ratio 10,000 : 15,000 simplifies to 2 : 3, which matches the capital time ratio 80,000 : 1,20,000. This confirms that the distribution is correct and consistent with the partnership principle.

Why Other Options Are Wrong:
The pairs 15,000 and 10,000, 5000 and 20,000, and 20,000 and 5000 either reverse the ratio or produce totals that do not match Rs. 25,000 when combined, or do not satisfy the required 2 : 3 ratio from the capital time products. Therefore they cannot represent the correct distribution of profit.

Common Pitfalls:
One common mistake is to think that equal capital means equal profit, ignoring the fact that one partner invested for a longer period. Another error is to divide the profit only on the basis of time without checking the actual ratio correctly. Always multiply capital by time for each partner first, simplify the resulting ratio, and only then split the profit.

Final Answer:
The respective shares of A and B are Rs. 10,000 and Rs. 15,000.