Das and Sinha are partners in a firm sharing profits in a $4:1$ ratio. They admit Pal as a new partner for a $\frac{1}{4}$ share in the profits, which he acquires entirely from Das. Calculate the new profit-sharing ratio of all three partners.

Let us calculate the new shares step by step:Identify the Old Shares:
$$\text{Das’s Old Share}=\frac{4}{5},\text{Sinha’s Old Share}=\frac{1}{5}$$
Pal's Admitted Share:
$$\text{Pal’s Share}=\frac{1}{4}$$
Calculate the New Shares After the Sacrifice:Since Pal acquires his share entirely from Das, we deduct $\frac{1}{4}$ only from Das's share:Das's New Share:
$$\text{Das’s New Share}=\frac{4}{5}-\frac{1}{4}=\frac{16-5}{20}=\frac{11}{20}$$
Sinha's New Share:Sinha's share remains unchanged, but we express it with a common denominator of 20:
$$\text{Sinha’s Share}=\frac{1}{5}=\frac{1\times 4}{5\times 4}=\frac{4}{20}$$
Pal's New Share:Expressing Pal's share with a common denominator of 20:
$$\text{Pal’s Share}=\frac{1}{4}=\frac{1\times 5}{4\times 5}=\frac{5}{20}$$
Combine into the New Ratio:Comparing their shares ($\frac{11}{20}:\frac{4}{20}:\frac{5}{20}$), the new profit-sharing ratio is $11:4:5$.