A, B and C are three equal partners. D is admitted for $${\frac{1}{4}^{{\text{th}}}}$$ share. What is the sacrificing ratio?

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

The sacrificing ratio is the ratio in which the old partners give up their share to the new partner. In this case, A, B, and C are three equal partners. When D is admitted for $\frac{1}{4}$ share, the old partners’ shares will be reduced by $\frac{1}{4}$ each. Therefore, the sacrificing ratio is $1:1:1$.

Here is a more detailed explanation of each option:

• Option A: $1:1:1$. This is the correct answer. The old partners give up $\frac{1}{4}$ share each, so the sacrificing ratio is $1:1:1$.
• Option B: $1:2:1$. This is incorrect. The old partners give up $\frac{1}{4}$ share each, so the sacrificing ratio cannot be $1:2:1$.
• Option C: $2:1:1$. This is incorrect. The old partners give up $\frac{1}{4}$ share each, so the sacrificing ratio cannot be $2:1:1$.
• Option D: $1:1:2$. This is incorrect. The old partners give up $\frac{1}{4}$ share each, so the sacrificing ratio cannot be $1:1:2$.