A and B start a business jointly. A invests Rs. 16,000 for 8 months. B remains in the business for 4 months and claims 2/7 of the total profit. How much capital did B contribute to the business?

Introduction / Context:
This is a partnership question in which two partners invest different amounts for different time periods. One partner, B, claims a known fraction of the total profit. Using the concept that profit is shared in proportion to capital multiplied by time, we must determine how much money B originally invested in order to justify a 2/7 share of the total profit.

Given Data / Assumptions:

• Partner A invests Rs. 16,000.
• A keeps this capital invested for 8 months.
• Partner B remains in the business for 4 months.
• B claims 2/7 of the total profit.
• Profit is shared in proportion to capital * time (effective investment).
• We are required to find the amount of capital contributed by B.

Concept / Approach:
For partnership problems, effective investment is calculated as capital * time. If the total profit is divided in a certain fraction for one partner, that fraction must match the ratio of that partner's effective investment to the total effective investment of all partners. Here, we let B's capital be an unknown, compute the ratio of effective investments and equate B's profit fraction to 2/7 to solve for B's capital.

Step-by-Step Solution:
Step 1: Effective investment of A = 16,000 * 8 = 128,000. Step 2: Let B's capital be y rupees. B is in the business for 4 months, so effective investment of B = y * 4 = 4y. Step 3: Total effective investment = 128,000 + 4y. Step 4: B's share of profit as a fraction of the total profit = effective investment of B / total effective investment = 4y / (128,000 + 4y). Step 5: According to the question, B's share of the profit is 2/7. So we set 4y / (128,000 + 4y) = 2 / 7. Step 6: Cross multiply to solve for y: 7 * 4y = 2 * (128,000 + 4y). Step 7: This gives 28y = 256,000 + 8y. Rearranging, 28y - 8y = 256,000 so 20y = 256,000. Step 8: Divide both sides by 20 to get y = 256,000 / 20 = 12,800. Step 9: Therefore, B's capital contribution is Rs. 12,800.

Verification / Alternative check:
Using A's effective investment of 128,000 and B's effective investment of 4 * 12,800 = 51,200, the total effective investment is 128,000 + 51,200 = 179,200. B's share fraction is then 51,200 / 179,200. Simplifying, divide numerator and denominator by 25,600 to get 2 / 7. This exactly matches the given profit share fraction, confirming that the value of B's capital is correct.

Why Other Options Are Wrong:
If B had invested Rs. 12,000, 13,000 or 14,500, the resulting effective investment ratios would give profit shares different from 2/7. For example, with Rs. 12,000, the fraction becomes 48,000 / (128,000 + 48,000) = 48,000 / 176,000 which simplifies to 3 / 11, not 2 / 7. Therefore these alternatives do not satisfy the condition on B's profit share.

Common Pitfalls:
Learners may confuse the fraction of investment with the fraction of profit or may ignore the difference in time for which each partner stays in the business. Another common mistake is forming the ratio 16,000 : y directly without multiplying by time periods. Always remember that partnership profit shares depend on capital * time, not just on capital alone when the durations are different.

Final Answer:
The capital contributed by B is Rs. 12,800.