A, B, C and D enter into a partnership. A subscribes one-third of the total capital, B subscribes one-fourth, C subscribes one-fifth and D contributes the remaining amount. If the total profit at the end of the year is Rs. 2490, what is A's share in the profit?

Introduction / Context:
This partnership question involves four partners contributing fixed fractions of the total capital. Since all partners are assumed to invest for the same duration, the profit is directly proportional to the fraction of capital contributed by each. The task is to compute A's share in a known total profit when the fractional contributions are given.

Given Data / Assumptions:

• A contributes one-third of the total capital.
• B contributes one-fourth of the total capital.
• C contributes one-fifth of the total capital.
• D contributes the remaining amount of capital.
• Total profit at the end of the year is Rs. 2490.
• All partners keep their money invested for the whole year.
• Profit is shared in proportion to the capital invested.

Concept / Approach:
Since all partners invest for the same time period, we can work purely with capital fractions. We first find D's fractional share by subtracting the fractions of A, B and C from 1. Then we convert these fractions into a simple ratio, which represents how the profit is split. Once we know the total number of ratio parts and the total profit, we can find the value of one part and hence A's share.

Step-by-Step Solution:
Step 1: A's fraction of capital is 1/3, B's is 1/4 and C's is 1/5. Step 2: Sum these fractions: 1/3 + 1/4 + 1/5. Step 3: The common denominator of 3, 4 and 5 is 60. So 1/3 = 20/60, 1/4 = 15/60 and 1/5 = 12/60. Step 4: Add them: 20/60 + 15/60 + 12/60 = 47/60. Step 5: Therefore, D's fraction is 1 - 47/60 = 13/60. Step 6: So the capital contributions are in the ratio A : B : C : D = 1/3 : 1/4 : 1/5 : 13/60. Step 7: Convert each fraction to a common denominator of 60 to express the ratio in whole numbers. Step 8: A = 1/3 = 20/60, B = 1/4 = 15/60, C = 1/5 = 12/60 and D = 13/60. Hence the ratio is 20 : 15 : 12 : 13. Step 9: Sum the parts: 20 + 15 + 12 + 13 = 60 parts. Step 10: Total profit is Rs. 2490, so one part is 2490 / 60 = Rs. 41.5. Step 11: A's share is 20 parts, so A gets 20 * 41.5 = Rs. 830.

Verification / Alternative check:
Check the other shares for consistency. B's share is 15 * 41.5 = Rs. 622.5, C's share is 12 * 41.5 = Rs. 498 and D's share is 13 * 41.5 = Rs. 539.5. The sum of these four amounts is 830 + 622.5 + 498 + 539.5 = Rs. 2490, which matches the total profit. This confirms that the ratio and the computed value for A's share are correct.

Why Other Options Are Wrong:
The other options such as Rs. 820, Rs. 840, Rs. 850 and Rs. 780 would require a different number of parts or a different value per part, which would then cause the sum of all four partners' shares to deviate from Rs. 2490. Only Rs. 830 is consistent with both the fraction based ratio and the given total profit.

Common Pitfalls:
A common error is to assume that D's share is just another simple fraction given directly instead of calculating it from the remaining portion of capital. Another frequent mistake is in adding the fractions incorrectly or in converting them to a common denominator. Some learners also forget to check that all four profit shares add to the total profit, which is a vital consistency check in partnership problems.

Final Answer:
A's share in the profit is Rs. 830, which corresponds to option B.