New Profit Sharing and Sacrificing Ratio (Part A)

Introduction:
R and T are partners in a firm sharing profits in the ratio of 3:2. S joins the firm and both R and T surrender a fraction of their shares in favor of S. The question requires us to find the new profit sharing and sacrificing ratio of the partners.

Calculation of the new profit sharing ratio:
The first step is to calculate the share of S in the profits of the firm. To do this, we need to know the total profit of the firm. Let us assume that the total profit of the firm is $X.

R's share = 3/5 * X
T's share = 2/5 * X
Total share of R and T = 3/5 * X + 2/5 * X = X

Let us assume that S is to receive y% of the total profit of the firm. Therefore, S's share = y/100 * X.

Now, R surrenders 1/4 of his share in favor of S. This means that R will now have (3/5 - 1/4) * X = 11/20 * X

Similarly, T surrenders 1/5 of his share in favor of S. This means that T will now have (2/5 - 1/5) * X = 1/5 * X.

Therefore, the new profit sharing ratio of R, T, and S will be 11:4:y.

Calculation of the sacrificing ratio:
The sacrificing ratio is the ratio in which the existing partners surrender their share in favor of the new partner. In this case, R and T are the existing partners, and S is the new partner. Therefore, we need to calculate the ratio in which R and T surrender their shares in favor of S.

R surrenders 1/4 of his share in favor of S. Therefore, the amount surrendered by R is (1/4) * (3/5) * X = 3/20 * X.

T surrenders 1/5 of his share in favor of S. Therefore, the amount surrendered by T is (1/5) * (2/5) * X = 2/25 * X.

Therefore, the sacrificing ratio of R and T in favor of S will be (3/20):(2/25) = 15:8.

Conclusion:
The new profit sharing and sacrificing ratio of the partners are 11:4:y and 15:8, respectively.