**New Profit Sharing Ratio**

To determine the new profit sharing ratio after C joins A and B as partners, we need to calculate the proportions of profit that each partner will receive based on the given information.

Let's assume the initial profit sharing ratio between A and B is x:y, where x represents A's share and y represents B's share. We need to find the new profit sharing ratio between A, B, and C.

**Calculation:**

1. A's share after C joins:
- A receives half of B's share, which means A will receive (1/2)y.
- Adding this to A's initial share, the new share for A becomes x + (1/2)y.

2. C's share after joining:
- C receives one third of A's share, which means C will receive (1/3)(x + (1/2)y).

3. B's share after C joins:
- B's share is the remaining portion of the profit after A and C have received their shares.
- It can be calculated by subtracting A's and C's shares from the total profit.
- B's share would be the total profit minus A's share and C's share, which is (1 - (x + (1/2)y + (1/3)(x + (1/2)y)).

**Simplifying the Calculation:**

To simplify the calculation, let's find a common denominator for (1/2)y and (1/3)(x + (1/2)y).

1. (1/2)y can be rewritten as (3/6)y.
2. (1/3)(x + (1/2)y) can be rewritten as (2/6)(x + (1/2)y).

Now, we can simplify the calculation as follows:

1. A's new share = x + (3/6)y
2. C's share = (2/6)(x + (1/2)y)
3. B's new share = 1 - (x + (3/6)y + (2/6)(x + (1/2)y))

**Final Profit Sharing Ratio:**

The new profit sharing ratio can be represented as:

A : B : C

(x + (3/6)y) : [1 - (x + (3/6)y + (2/6)(x + (1/2)y))] : (2/6)(x + (1/2)y)

Simplifying this further, we can express the new profit sharing ratio as:

(6x + 3y) : [6 - (3x + 2y)] : (2x + y)

Therefore, the new profit sharing ratio between A, B, and C is (6x + 3y) : [6 - (3x + 2y)] : (2x + y).