A and B are partners sharing profits and losses in 3 : 2. They admitted C in the firm. C acquired $${\frac{1}{3}^{{\text{rd}}}}$$ of A’s shares and $${\frac{1}{2}^{{\text{nd}}}}$$ of B’s share. What is the new profit sharing ratio?

The correct answer is $\boxed{\text{D}}$.

A and B share profits and losses in the ratio of 3:2. This means that A gets 3 parts of the profit for every 2 parts that B gets. When C is admitted to the firm, he acquires $\frac{1}{3}$ of A’s shares and $\frac{1}{2}$ of B’s shares. This means that A now gets 2 parts of the profit for every 3 parts that B gets and C gets 1 part of the profit for every 3 parts that A and B get together. Therefore, the new profit sharing ratio is 3:2:2.

Option A is incorrect because it gives C a share of 5 parts of the profit for every 3 parts that A and B get together. This is not possible because the total number of parts must always be equal to the number of partners.

Option B is incorrect because it gives C a share of 2 parts of the profit for every 1 part that A gets. This is not possible because A’s share must always be at least 1 part.

Option C is incorrect because it gives C a share of 1 part of the profit for every 2 parts that A and B get together. This is not possible because the total number of parts must always be equal to the number of partners.