A starts a business with ₹9000. B joins after 6 months with ₹45000. Using capital–time (capital-months), what is the ratio of profits of A and B at the end of the year?

Introduction / Context:
Different joining times require time-weighted capital. Compute capital-months for both partners and then form the profit ratio A : B accordingly.

Given Data / Assumptions:

• A: ₹9000 for 12 months.
• B: ₹45000 for 6 months (joined after 6 months).
• No salary, no interest, profits proportional to capital-months.

Concept / Approach:
Profit share ∝ capital × time. Multiply and reduce the ratio to simplest terms.

Step-by-Step Solution:
A’s capital-months = 9000 * 12 = 108000B’s capital-months = 45000 * 6 = 270000Ratio A : B = 108000 : 270000 = 2 : 5

Verification / Alternative check:
Divide both by 54000 to confirm 2 : 5.

Why Other Options Are Wrong:
1 : 5 or 5 : 2 do not match the computed capital-months. 5 : 1 reverses the true dominance of B’s contribution.

Common Pitfalls:
Using only capital (9k vs 45k) and forgetting time, which would give the wrong split.

Final Answer:
2 : 5