Solution:

Given:

- A, B, and C are partners sharing ratio 4:3:2
- D is admitted for 1/9 share
- A retains his original share

To find:

- New profit sharing ratio
- New sacrificing ratio

Calculation of New Profit Sharing Ratio:

- Initially, the total ratio of the partners was 4+3+2 = 9
- A's share = 4/9 of the total profit
- B's share = 3/9 of the total profit
- C's share = 2/9 of the total profit
- D's share = 1/9 of the total profit (as per the agreement)

Now, let's calculate the new profit sharing ratio after the admission of D:

- A's share = 4/9 (as he retains his original share)
- B's share = 3/9 * (8/9) = 2/9
- Here, we multiplied B's share by (8/9) because D's share is 1/9 and the remaining share is 8/9. So, B's share in the remaining 8/9 would be (3/9 * 8/9) = 2/9.
- C's share = 2/9 * (8/9) = 16/81
- Here, we multiplied C's share by (8/9) because D's share is 1/9 and the remaining share is 8/9. So, C's share in the remaining 8/9 would be (2/9 * 8/9) = 16/81.
- D's share = 1/9

Therefore, the new profit sharing ratio would be:

- A:B:C:D = 4/9:2/9:16/81:1/9
- Simplifying the ratio by multiplying each term by 81, we get:
- A:B:C:D = 32:16:16:9

Calculation of New Sacrificing Ratio:

- Sacrificing ratio is the ratio in which the partners agree to give up their share of profit for the admission of a new partner.
- Here, A retains his original share. So, the sacrificing ratio would be the ratio in which B and C agree to give up their share of profit for the admission of D.
- Let's assume that B and C agree to give up x and y shares of profit, respectively.
- Total profit share = 1
- As per the agreement, D is admitted for 1/9 share. So, the remaining share would be (1-1/9) = 8/9.
- The new profit sharing ratio between B, C, and D is 2:16:9.
- Therefore, the sacrificing ratio between B and C would be in the ratio of their profit share in the new ratio.
- Sacrificing ratio of B:C = (16/27):(9/27) = 16:9

Hence, the new profit sharing ratio and sacrificing ratio are:

- Profit sharing ratio: A:B:C:D = 32:16:16:9
- Sacrificing ratio: B:C = 16:9

D's share = 1/9
remaining shares = 1-1/9
=8/9
A's new share = 4/9×8/9
= 32/81
B's new share = 3/9×8/9
=24/81
C's new share = 2/9×8/9
= 16/81
D's share =1/9×9/9
= 9/81
hence the new profit sharing ratio is 32:24:16:9