In a partnership, the capitals of A, B, and C are such that 4 times A's capital equals 6 times B's capital equals 10 times C's capital. If the total profit is Rs. 4650, how much of this profit will C receive?

Introduction / Context:
This partnership question gives a relationship between the capitals of three partners in the form of equal products, namely 4 times A's capital, 6 times B's capital, and 10 times C's capital are all equal. From this information, we must determine the ratio of their capitals and hence the ratio in which they share the profit. Finally, we use this ratio to find C's share of the total profit of Rs. 4650.

Given Data / Assumptions:
- 4 times A's capital equals 6 times B's capital equals 10 times C's capital.
- Total profit to be shared is Rs. 4650.
- The partners share profit in direct proportion to their capitals, and all capitals are assumed to be invested for the same time.

Concept / Approach:
Let the common value of 4A, 6B, and 10C be k. That is, 4A = 6B = 10C = k. From this, we can express A, B, and C in terms of k and then find the ratio A : B : C. Once we obtain this ratio, we calculate how many parts belong to C and what each part of the total profit is worth. Multiplying the value of one part by C's number of parts gives C's profit share.

Step-by-Step Solution:
Step 1: Let 4A = 6B = 10C = k.Step 2: Then A = k / 4, B = k / 6, and C = k / 10.Step 3: The ratio A : B : C is (k / 4) : (k / 6) : (k / 10). The k cancels out, leaving 1/4 : 1/6 : 1/10.Step 4: To clear denominators, multiply each term by the least common multiple of 4, 6, and 10, which is 60. So we get A : B : C = 60 * (1/4) : 60 * (1/6) : 60 * (1/10) = 15 : 10 : 6.Step 5: Total number of parts = 15 + 10 + 6 = 31.Step 6: Each part of profit = total profit / 31 = 4650 / 31. Since 31 * 150 = 4650, each part is worth Rs. 150.Step 7: C has 6 parts, so C's share = 6 * 150 = Rs. 900.

Verification / Alternative check:
We can also compute A's and B's shares to verify the total. A receives 15 * 150 = Rs. 2250. B receives 10 * 150 = Rs. 1500. C receives 6 * 150 = Rs. 900. Adding them gives 2250 + 1500 + 900 = Rs. 4650, which matches the total profit. This confirms that the distribution is correct and that C's share is indeed Rs. 900.

Why Other Options Are Wrong:
If C received Rs. 700 or Rs. 800, the implied value of each part would not divide Rs. 4650 evenly according to the 15 : 10 : 6 ratio. A share of Rs. 1000 would also not maintain the correct ratio and would cause the total of all three shares to differ from Rs. 4650. Only Rs. 900 leads to a consistent distribution across all three partners.

Common Pitfalls:
Some candidates misinterpret 4A = 6B = 10C and attempt to write the ratio directly as 4 : 6 : 10, which is incorrect. The correct method is to express A, B, and C in terms of the common value and then convert them into a simple integer ratio. Another mistake is to forget to add all parts together when computing the value of one part. Working carefully through each algebraic step avoids these errors.

Final Answer:
C will receive a profit share of Rs. 900.