A and B are partners in a firm sharing profits in the ratio of 3: 2, C is admitted for $\frac{3}{20}$ share in profits. C aquires $\frac{1}{3}$ of his share from B and A surrenders $\frac{1}{6}$th of his share to C. What will be the new ratio of the firm?

Question

Accepted Answer

The correct option is 1: (10:7:3)

The new profit-sharing ratio is determined by subtracting the sacrifice made by the existing partners (A and B) from their original shares and including the new partner's (C's) share. First, the amount of sacrifice by each partner must be calculated. The old ratio of A and B is 3:2, meaning A's original share is $\frac{3}{5}$ and B's is $\frac{2}{5}$. A surrenders $\frac{1}{6}$th of his share, so A's sacrifice is $\frac{1}{6} \times \frac{3}{5} = \frac{1}{10}$. B's sacrifice is calculated based on C's total share ($\frac{3}{20}$), as C acquires $\frac{1}{3}$ *of his share* from B. Thus, B's sacrifice is $\frac{1}{3} \times \frac{3}{20} = \frac{1}{20}$. Now, we calculate the new shares. A's new share is $\frac{3}{5} - \frac{1}{10} = \frac{6-1}{10} = \frac{5}{10}$. B's new share is $\frac{2}{5} - \frac{1}{20} = \frac{8-1}{20} = \frac{7}{20}$. C's share is $\frac{3}{20}$. Expressing the new ratio with a common denominator (20) gives: $\frac{10}{20} : \frac{7}{20} : \frac{3}{20}$, resulting in the new profit-sharing ratio of **10:7:3**.

The incorrect options are 2, 3, and 4:

Option 2 (15:08:07) is incorrect because it is mathematically inconsistent with the defined terms of surrender and acquisition. This result would likely stem from a fundamental error in calculating the fractions of sacrifice, such as incorrectly applying a fraction to the firm's total profit instead of the partner's individual share. Furthermore, a common mistake is misinterpreting the language "C acquires $1/3$ of his share from B" versus an outright surrender *by* B. The resulting ratio must ensure that the total of the new shares equals unity (1), which this ratio would not correctly achieve under the problem's conditions.

Option 3 (5:04:01) is incorrect as it represents a significant deviation from the correct proportional shares. The resulting ratio is much too small, suggesting an error where the partners' *remaining* shares after C's admission were used in a faulty calculation for the new ratio's base. It might arise if A's and B's sacrifices were calculated as an amount *from* their shares (e.g., $\frac{1}{6}$ and $\frac{1}{3}$ directly) instead of *of* their shares or *of* C's share, or if fractions were added instead of subtracted. In partnership accounting, precision in calculating the exact amount of sacrifice is vital for the new ratio to be correct.

Option 4 (None of these) is incorrect because the question provides complete and unambiguous data for the calculation of the new ratio. The shares surrendered by A ($\frac{1}{10}$) and B ($\frac{1}{20}$) correctly sum up to C's total acquired share ($\frac{3}{20}$), which confirms the data's consistency. Since a clear, mathematically verifiable ratio (10:7:3) exists, the choice "None of these" cannot be the correct answer. The new ratio must always be calculated to ensure that all partners' shares collectively amount to 1.