Deriving time ratio from given capital and profit ratios: A, B, and C invested in the ratio 4 : 6 : 9 and received profit in the ratio 2 : 3 : 5. Find the ratio of time for which their capitals were invested.

Introduction / Context:
Profit is proportional to capital × time. With known capital and profit ratios, we can recover time parts by dividing profit parts by capital parts for each partner. This inverse approach is essential for handling varying durations of investment in partnerships.

Given Data / Assumptions:

• Capital ratio (A : B : C) = 4 : 6 : 9.
• Profit ratio (A : B : C) = 2 : 3 : 5.
• Time ∝ profit / capital.

Concept / Approach:
Compute time parts: t_A ∝ 2/4, t_B ∝ 3/6, t_C ∝ 5/9. Convert to a whole-number ratio by clearing denominators through a common multiple (here, 18).

Step-by-Step Solution:

Verification / Alternative check:
Multiply capital by time: A 4*9=36, B 6*9=54, C 9*10=90 ⇒ 36:54:90 reduces to 2:3:5, confirming the profit ratio.

Why Other Options Are Wrong:
Other choices do not reproduce the given profit ratio when combined with the capital ratio via capital × time.

Common Pitfalls:
Using capital/time instead of profit/capital, or forgetting to scale to clear denominators properly.

Final Answer:
9 : 9 : 10