K and L start a business jointly. K invests Rs. 16,000 for 8 months. L remains in the business for only 4 months, and out of the total profit L claims 2/7 of the share. How much money did L contribute to the business?

Introduction / Context:
This partnership question asks you to find the capital contributed by one partner when their share in profit and the time periods of investment are known. The key relationship is that the share of profit is proportional to the product of capital and time for each partner.

Given Data / Assumptions:

• K invests Rs. 16,000 for 8 months.
• L invests an unknown amount for 4 months.
• L gets 2/7 of the total profit and K gets the remaining 5/7.
• Profit is shared in proportion to capital time products.
• We must determine L's invested capital.

Concept / Approach:
Let K's and L's capital time products be CK and CL. Their profit shares must satisfy CK : CL = K's share : L's share. Since we know the fraction of total profit going to L, we can set up a ratio for the capital time products. From this ratio, we solve for the unknown capital contributed by L, using the known time periods for each partner.

Step-by-Step Solution:
Step 1: Compute K's capital time product: CK = 16,000 * 8 = 1,28,000. Step 2: Let L's investment be x rupees for 4 months. Step 3: L's capital time product: CL = x * 4 = 4x. Step 4: L receives 2/7 of the profit, so K must receive 5/7 of the profit. Step 5: Therefore the ratio CK : CL = K's share : L's share = 5 : 2. Step 6: Set up the proportion: 1,28,000 : 4x = 5 : 2. Step 7: Cross multiply: 1,28,000 * 2 = 5 * 4x, giving 2,56,000 = 20x. Step 8: Solve for x: x = 2,56,000 / 20 = 12,800. Step 9: Hence, L contributed Rs. 12,800 to the business.

Verification / Alternative check:
With L's capital equal to Rs. 12,800, his capital time product is 12,800 * 4 = 51,200. K's product is 1,28,000. The ratio CK : CL = 1,28,000 : 51,200. Dividing both numbers by 25,600 gives 5 : 2, which matches the ratio of profit shares K : L. This confirms that L receiving 2/7 of the profit is consistent with his capital time product and that our computed investment is correct.

Why Other Options Are Wrong:
Rs. 13,204, Rs. 14,521 and Rs. 15,000 produce capital time products that do not give the exact 5 : 2 ratio when compared to K's product of 1,28,000. Using these values leads to profit shares that do not match the stated 2/7 and 5/7 split, so they cannot be correct.

Common Pitfalls:
A typical mistake is to treat the share 2/7 as the ratio of capitals directly without considering the different investment durations. Another error is to set up the ratio as 2 : 5 instead of 5 : 2, which inverts the relationship and produces the reciprocal answer. Always link the share of profit to the product of capital and time, not to capital alone, and double check the order of terms in the ratio.

Final Answer:
L contributed Rs. 12,800 to the business.