Introduction:
When a new partner is admitted to a partnership firm, it is essential to determine the new profit sharing ratio among the partners. This ratio determines how the profits and losses of the firm will be distributed among the partners.

Given Information:
- A and B are partners sharing in the ratio of 3:2.
- C is admitted for a 3/10th share, which he acquires in the ratio of 2:1 from the old partners.

Calculating C's Share:
To determine C's share, we need to find out the value of 3/10th share and then divide it in the ratio of 2:1 between A and B.

Step 1: Calculate the total share of A and B.
The total share is the sum of the ratios:
3 + 2 = 5

Step 2: Calculate C's share.
C's share is 3/10th of the total share.
C's share = (3/10) * 5 = 3/2

Step 3: Divide C's share between A and B.
The ratio of C's share between A and B is 2:1.
A's share = (2/3) * (3/2) = 1
B's share = (1/3) * (3/2) = 1/2

Therefore, the new profit sharing ratio among A, B, and C is 1:1/2:1.

Explanation:
When a new partner is admitted to a partnership firm, the existing partners need to agree on the new profit sharing ratio. This ratio determines how the profits and losses of the firm will be distributed among the partners.

In this case, A and B are the old partners sharing in the ratio of 3:2. This means that out of the total profits, A gets 3 parts and B gets 2 parts.

C is admitted for a 3/10th share, which means that C will receive 3/10th of the total profits. However, this share needs to be divided between A and B in the ratio of 2:1.

To calculate C's share, we first find out the total share of A and B, which is the sum of their ratios (3 + 2 = 5). Then, we calculate C's share as 3/10th of the total share.

Finally, we divide C's share between A and B in the ratio of 2:1. This means that A gets 2 parts out of 3 and B gets 1 part out of 3.

Therefore, the new profit sharing ratio among A, B, and C is 1:1/2:1. This ratio determines how the profits and losses of the firm will be distributed among the partners.