A, B, and C start a business with initial investments in the ratio 1 : 3 : 5. After 4 months, A invests the same amount again as he had originally invested, while B and C each withdraw half of their original capital. If the business runs for one year in total, in what ratio will the profits of A, B, and C be shared at the end of the year?

Introduction / Context:
This question is a classic example of a partnership problem with changing investments over time. Instead of keeping capitals fixed, one partner increases his investment and the other partners reduce theirs after a certain period. Profit in such problems is shared in proportion to the product of capital and time, often called capital time. Understanding this concept is crucial for solving variable partnership questions in competitive exams.

Given Data / Assumptions:
- Initial ratio of investments of A, B, and C is 1 : 3 : 5.
- After 4 months, A invests the same amount again as before, so his capital doubles from that point onward.
- After 4 months, B withdraws half of his capital and C also withdraws half of his capital.
- The total duration of the business is 12 months (1 year).
- Profit is shared according to capital multiplied by the number of months for which that capital is used.

Concept / Approach:
For each partner, we calculate capital time in two segments. For the first 4 months, all three partners use their original capitals in the ratio 1 : 3 : 5. For the remaining 8 months, A uses double his original capital, while B and C each use half of their respective original capitals. We then add capital time from both segments for each partner and convert the final values into a simplified ratio to obtain the profit sharing ratio.

Step-by-Step Solution:
Step 1: Let the original capitals be A = 1 unit, B = 3 units, and C = 5 units.Step 2: For the first 4 months, capital time is: A = 1 * 4 = 4, B = 3 * 4 = 12, C = 5 * 4 = 20.Step 3: After 4 months, A doubles his capital to 2 units. B and C reduce to half, so B = 1.5 units and C = 2.5 units.Step 4: For the remaining 8 months, capital time is: A = 2 * 8 = 16, B = 1.5 * 8 = 12, C = 2.5 * 8 = 20.Step 5: Add both segments. A's total capital time = 4 + 16 = 20, B's total = 12 + 12 = 24, C's total = 20 + 20 = 40.Step 6: The ratio 20 : 24 : 40 simplifies by dividing by 4 to 5 : 6 : 10.

Verification / Alternative check:
We can quickly verify by rechecking computations. First 4 months: 1, 3, and 5 units are correct. Next 8 months: doubling 1 gives 2, half of 3 gives 1.5, and half of 5 gives 2.5, which are consistent. The capital time calculations match, and simplifying 20 : 24 : 40 by 4 indeed gives 5 : 6 : 10. Therefore the profit ratio is confirmed.

Why Other Options Are Wrong:
The ratio 1 : 2 : 3 does not reflect the larger long term investment of C and the increased later investment of A. The option 3 : 4 : 15 gives C a disproportionately large share that is not supported by the computed capital times. The ratio 3 : 5 : 10 also does not align with the calculated 20 : 24 : 40. Only 5 : 6 : 10 matches the actual capital time distribution.

Common Pitfalls:
Many learners mistakenly use only the initial capital ratio and ignore the change after 4 months, or they forget that the profit share depends on both capital and time. Some also incorrectly assume that A invests double for the entire year rather than only for the last 8 months. Another frequent mistake is to miscalculate 1.5 * 8 or 2.5 * 8, which can distort the final ratio. Always handle each time segment separately and then combine them carefully.

Final Answer:
The profits of A, B, and C will be shared in the ratio 5 : 6 : 10.