Profit ratio when a third partner joins mid-year: A and B invest in the ratio 3 : 5 for the entire year. After 6 months, C joins with the same capital as B for the remaining 6 months. Find the profit ratio A : B : C at year end.

Introduction / Context:
When partners enter at different times or with different amounts, profit division uses capital × time. A and B invest for 12 months, while C enters after 6 months with a specified capital. We convert each contribution to equivalent “money-months” and then form the ratio of these contributions to split the profits.

Given Data / Assumptions:

• A : B (capital) = 3 : 5 for 12 months each.
• C’s capital equals B’s capital and is invested for 6 months.
• Profit share ∝ capital * time.

Concept / Approach:
Introduce a base unit k for capital. A invests 3k for 12 months; B invests 5k for 12 months; C invests 5k for 6 months. Compute weights and reduce the resulting ratio to simplest form.

Step-by-Step Solution:

Verification / Alternative check:
If total profit were P, shares would be proportional to 6, 10, and 5 parts. Any consistent P will preserve this ratio across A, B, C.

Why Other Options Are Wrong:
3 : 5 : 5 and 3 : 5 : 2 ignore time differences; 6 : 2 : 3 misplaces weights; 4 : 5 : 3 does not match the computed 36 : 60 : 30 basis.

Common Pitfalls:
Confusing “equal to B’s capital” with “equal time”; forgetting that C invests only for 6 months; skipping the reduction step of the ratio correctly.

Final Answer:
6 : 10 : 5