A and B are partners in a firm sharing profit and loss in the ratio of 3 : 2. They admit C into partnership for $${\frac{1}{4}^{{\text{th}}}}$$ share and the new ratio between A and B is 2 : 1. The sacrificing ratio is

The correct answer is $\boxed{\text{B. 2 : 1}}$.

Let $x$ be the amount that A and B originally invested in the business. Then, A’s share of the profit was $\frac{3}{5}x$ and B’s share was $\frac{2}{5}x$. When C is admitted, the new total share is $1$ share. Therefore, A and B must give up a total of $\frac{1}{4}$ share to C. This means that A gives up $\frac{3}{10}$ share and B gives up $\frac{2}{10}$ share. The sacrificing ratio is the ratio of the shares given up by A and B, which is $\frac{3}{10} : \frac{2}{10} = 3 : 2$.

Here is a step-by-step solution:

1. Let $x$ be the amount that A and B originally invested in the business.
2. A’s share of the profit was $\frac{3}{5}x$ and B’s share was $\frac{2}{5}x$.
3. When C is admitted, the new total share is $1$ share.
4. Therefore, A and B must give up a total of $\frac{1}{4}$ share to C.
5. This means that A gives up $\frac{3}{10}$ share and B gives up $\frac{2}{10}$ share.
6. The sacrificing ratio is the ratio of the shares given up by A and B, which is $\frac{3}{10} : \frac{2}{10} = 3 : 2$.