A, B, and C invest for time periods in the ratio 2 : 1 : 8. Their profits at the end are in the ratio 3 : 4 : 2. Find the ratio of their investments (capitals) A : B : C.

Introduction / Context:
Profit ∝ capital * time. With time ratios given and profit ratios known, you can back out the capital ratio by dividing profit ratios by time ratios termwise.

Given Data / Assumptions:
Time ratio = 2 : 1 : 8; Profit ratio = 3 : 4 : 2.

Concept / Approach:
For each partner, capital ∝ profit / time. Apply this to all three and then simplify the results to obtain A : B : C.

Step-by-Step Solution:
A's capital ∝ 3 / 2 = 1.5. B's capital ∝ 4 / 1 = 4. C's capital ∝ 2 / 8 = 0.25. To clear decimals, multiply all by 4 ⇒ A : B : C = 6 : 16 : 1.

Verification / Alternative check:
Using these capitals with the given times: (6*2):(16*1):(1*8) = 12:16:8 = 3:4:2, which matches the profit ratio.

Why Other Options Are Wrong:
They do not reproduce the 3:4:2 profit ratio when multiplied by the time ratio 2:1:8.

Common Pitfalls:
Multiplying profit and time instead of dividing, or forgetting to scale to remove fractions.

Final Answer:
6 : 16 : 1