A starts a business with ₹4000. B joins after 3 months with ₹16000. At the end of one year, what is the ratio of their profits (A : B) based on capital–time?

Introduction / Context:
When partners join at different times, the correct split uses capital × time (capital-months). Compute each partner’s capital-months and take the ratio.

Given Data / Assumptions:

• A: ₹4000 invested for 12 months.
• B: ₹16000 invested for 9 months (joined after 3 months).
• No other adjustments such as drawings or salaries.

Concept / Approach:
Profit share ∝ capital × time. Multiply the capital by months of use, then form their ratio and reduce to simplest terms.

Step-by-Step Solution:
A’s capital-months = 4000 * 12 = 48000B’s capital-months = 16000 * 9 = 144000Ratio A : B = 48000 : 144000 = 1 : 3

Verification / Alternative check:
Divide both by 48000 to confirm the simplified ratio 1 : 3.

Why Other Options Are Wrong:
2 : 3 and others do not reflect the much larger capital-months contributed by B.

Common Pitfalls:
Ignoring the late entry and treating times equally; this would incorrectly produce a 1 : 4 capital-only ratio instead of the correct capital–time ratio.

Final Answer:
1 : 3