A, B and C enter into a partnership. A contributes Rs. 10,000 as capital. Out of a total profit of Rs. 1,000, A receives Rs. 500 and B receives Rs. 300. Based on this information, what is the amount of capital invested by C?

Introduction / Context:
This question is about partnership and capital contribution. Three partners A, B and C share a known profit in known amounts. We are told A's capital investment but not the capitals of B and C. Using the relationship between profit shares and capital contributions, we must determine how much capital C invested in the business.

Given Data / Assumptions:

• A, B and C are partners in a business.
• A invests Rs. 10,000 as capital.
• Total profit at the end of the period is Rs. 1,000.
• Out of this profit, A gets Rs. 500.
• B gets Rs. 300 from the profit.
• Therefore C gets the remaining Rs. 200.
• The time period of investment is the same for all partners.
• Profit is shared in proportion to capital investments.

Concept / Approach:
In partnership questions where all partners keep their capital invested for the same time, the ratio of their profits is the same as the ratio of their capitals. We first write the profit shares of A, B and C as a ratio, then correlate this with their capitals. We know A's capital, so we scale the ratio to match A's capital and then read off C's capital from the same scale.

Step-by-Step Solution:
Step 1: Write the profit amounts: A = 500, B = 300 and C = 1,000 - (500 + 300) = 200. Step 2: Express these as a ratio: 500 : 300 : 200. Step 3: Simplify the ratio by dividing all terms by 100 to get 5 : 3 : 2. Step 4: Since the investment period is the same, capital ratio = profit ratio = 5 : 3 : 2. Step 5: A's capital corresponds to 5 parts and is given as Rs. 10,000. Step 6: Value of 1 part = 10,000 / 5 = Rs. 2,000. Step 7: C's capital corresponds to 2 parts, so C's capital = 2 * 2,000 = Rs. 4,000.

Verification / Alternative check:
Using the capitals in the ratio 10,000 : 6,000 : 4,000 (which matches 5 : 3 : 2), we can verify that the profit division matches the question. The total capital is 20,000. A's share would be 10,000 / 20,000 of the profit = 1/2 of 1,000 = 500. B's share would be 6,000 / 20,000 of the profit = 3/10 of 1,000 = 300. C's share would be 4,000 / 20,000 of the profit = 1/5 of 1,000 = 200. This matches the given distribution exactly, which confirms our calculation of C's capital.

Why Other Options Are Wrong:
If C had invested Rs. 5,000, Rs. 6,000 or Rs. 7,000, the capital ratio would not be consistent with the profit ratio 5 : 3 : 2. The resulting profit shares for such capitals would not match the given values of 500, 300 and 200, so those options cannot be correct.

Common Pitfalls:
A frequent mistake is to simply subtract amounts or to assume capital differences rather than ratios. Some learners also forget to compute C's profit share before forming the ratio. Always derive the complete profit ratio, simplify it and then match it to the known capital of one partner to scale everything correctly.

Final Answer:
The capital invested by C is Rs. 4,000.