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

The correct answer is $\boxed{\text{B. 1 : 4}}$.

To calculate the sacrificing ratio, we first need to calculate the total number of shares in the business after C is admitted. This is done by adding the original number of shares held by A and B to the new share held by C. In this case, the total number of shares is $2 + 3 + \frac{1}{4} = \frac{13}{4}$.

Next, we need to calculate the new share ratio of A and B. This is done by dividing the original number of shares held by each partner by the total number of shares. In this case, the new share ratio of A and B is $\frac{2}{\frac{13}{4}} = \frac{8}{13}$.

Finally, we can calculate the sacrificing ratio by subtracting the new share ratio from the original share ratio. In this case, the sacrificing ratio of A and B is $\frac{2}{3} – \frac{8}{13} = \boxed{\frac{1}{4}}$.

Here is a brief explanation of each option:

• Option A: $3 : 1$. This is the correct answer if the total number of shares in the business after C is admitted is 3. However, the total number of shares in the business is actually $\frac{13}{4}$.
• Option B: $1 : 4$. This is the correct answer.
• Option C: $2 : 3$. This is the original share ratio of A and B. However, we need to calculate the new share ratio of A and B after C is admitted.
• Option D: $1 : 1$. This is incorrect because it does not take into account the fact that C is admitted for a $\frac{1}{4}^{{\text{th}}}$ share in the business.