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Profit Sharing Ratio:
The profit sharing ratio is the proportion in which the partners distribute the profits among themselves. In this case, A, B, and C share the profits in the ratio of 5:3:2.

To calculate the new profit sharing ratio, we need to consider the changes in the partnership agreement.

Sacrificing Ratio:
The sacrificing ratio represents the proportion in which the existing partners sacrifice their share of profit in favor of the incoming partner(s). It is used when a new partner is admitted or an existing partner retires or dies.

To calculate the sacrificing ratio, we need to know the ratio in which the existing partners' share of profit is reduced.

Case 1: A Retires and B and C Continue:
When A retires, the remaining partners, B and C, will continue the business. Since A is retiring, he will sacrifice his share of profit in favor of B and C.

To calculate the sacrificing ratio, we need to find the difference between A's current profit sharing ratio and the new profit sharing ratio of B and C.

A's current profit sharing ratio = 5
B and C's current profit sharing ratio = 3+2 = 5

Sacrificing ratio = A's current profit sharing ratio - B and C's current profit sharing ratio
= 5 - 5
= 0

The sacrificing ratio in this case is 0, as A's share of profit is completely sacrificed.

The new profit sharing ratio of B and C will be the same as their current profit sharing ratio, which is 5:5 or 1:1.

Case 2: D Admitted and B and C Continue:
When a new partner, D, is admitted, the existing partners, B and C, will continue the business. The profit sharing ratio will change to accommodate the new partner.

To calculate the new profit sharing ratio, we need to consider the share of profit D will receive and the remaining share for B and C.

Let's assume D's share of profit is x. The new profit sharing ratio will be (B's share):(C's share):(D's share).

According to the given ratio, B's share = 3, C's share = 2.

Therefore, the new profit sharing ratio will be 3:2:x.

To find x, we can use the fact that the sum of the profit sharing ratios is equal to the total profit. In this case, the sum is 5+3+2 = 10.

So, 3+2+x = 10
x = 5

Therefore, the new profit sharing ratio will be 3:2:5.

Summary:
- In Case 1, where A retires and B and C continue, the sacrificing ratio is 0 and the new profit sharing ratio of B and C is 1:1.
- In Case 2, where D is admitted and B and C continue, the new profit sharing ratio is 3:2:5.