Finding time ratios from profit and capital ratios: A, B, C invested capitals in the ratio 5 : 6 : 8, but received profits in the ratio 5 : 3 : 1. Determine the ratio of time for which their capitals were invested.

Introduction / Context:
Profit in partnerships is proportional to capital × time. If we know both the capital ratio and the final profit ratio, we can back out the time ratio by dividing profit parts by capital parts for each partner. This technique is widely used to resolve differing investment durations.

Given Data / Assumptions:

• Capital ratio (A : B : C) = 5 : 6 : 8.
• Profit ratio (A : B : C) = 5 : 3 : 1.
• Profit ∝ capital * time ⇒ time ∝ profit / capital.

Concept / Approach:
Compute time ratios as (profit_i / capital_i) for each partner and then scale to whole numbers by a common factor to remove fractions.

Step-by-Step Solution:

Verification / Alternative check:
Check: capital × time ⇒ A 5*8=40, B 6*4=24, C 8*1=8 ⇒ profit parts 40:24:8 = 5:3:1 after dividing by 8, consistent with the given profit ratio.

Why Other Options Are Wrong:
Other options fail to produce the stated profit ratio when multiplied by capital parts; only 8 : 4 : 1 satisfies profit ∝ capital × time exactly.

Common Pitfalls:
Inverting the relationship (capital/time) or trying to subtract ratios instead of dividing; always use profit / capital to get time parts.

Final Answer:
8 : 4 : 1