The old Profit Sharing ratio among M, N and P are 2:2:1. The New profit sharing ratio after N retirement is 3:2. The gaining ratio between M and P will be:

Given Information:

Old profit sharing ratio: $M:N:P = 2:2:1$

N is retiring from the partnership

New profit sharing ratio: $M:P = 3:2$

Find: Gaining ratio between M and P

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When a partner retires, their share gets distributed among the remaining partners. The gaining ratio shows how much extra profit each remaining partner receives.

Think of it like this: N had a slice of the profit pie. Now that N is leaving, M and P will split that slice between them. The gaining ratio tells us how they split it.

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Old ratio $= 2:2:1$

Total parts $= 2 + 2 + 1 = 5$

M's old share $= \dfrac{2}{5}$

N's old share $= \dfrac{2}{5}$

P's old share $= \dfrac{1}{5}$

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New ratio $= 3:2$ (only M and P remain)

Total parts $= 3 + 2 = 5$

M's new share $= \dfrac{3}{5}$

P's new share $= \dfrac{2}{5}$

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Gain $=$ New Share $-$ Old Share

M's gain:

$\dfrac{3}{5} - \dfrac{2}{5} = \dfrac{1}{5}$

P's gain:

$\dfrac{2}{5} - \dfrac{1}{5} = \dfrac{1}{5}$

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Gaining Ratio $=$ M's gain $:$ P's gain

$= \dfrac{1}{5} : \dfrac{1}{5}$

Multiplying both sides by $5$:

$= 1 : 1$

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Verification: M and P should gain a total equal to N's old share.

Total gain $= \dfrac{1}{5} + \dfrac{1}{5} = \dfrac{2}{5}$

N's old share $= \dfrac{2}{5}$

The numbers match, confirming our answer.

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Final Answer: $1:1$