Question

A, B and C are partners sharing profits and losses in the ratio of 5 : 4 : 1. Calculate new profit-sharing ratio, sacrificing ratio and gaining ratio in each of the following cases:
Case 1. C acquires 1/5th share from A.
Case 2. C acquires 1/5th share equally form A and B.
Case 3. A, B and C will share future profits and losses equally.
Case 4. C acquires 1/10th share of A and 1/2 share of B.

Answer 1

$\begin{array}{l}\text{Calculation of New Profit Sharing Ratio}\\ \mathbf{\text{Case 1}}\text{:}\\ \text{A:B:C}=\text{5:4:1}\left(\text{Old Ratio}\right)\\ \text{C acquires}\frac{1}{5}\text{th from A}\\ \text{A's sacrifice =}\frac{1}{5}\\ \text{C's gain =}\frac{1}{5}\\ \text{A}=\frac{5}{10}-\frac{1}{5}=\frac{5-2}{10}=\frac{3}{10}\\ \text{B}=\frac{4}{10}\\ \text{C}=\frac{1}{10}+\frac{1}{5}=\frac{1+2}{10}=\frac{3}{10}\\ \text{A:B:C}=\text{3:4:3}\\ \mathbf{\text{Case 2}}\text{:}\\ \text{A:B:C}=\text{5:4:1}\left(\text{Old Ratio}\right)\\ \text{C acquires}\frac{1}{5}\text{th share equally from A}\\ \text{A's sacrifice =}\frac{1}{10}\\ \text{B's sacrifice =}\frac{1}{10}\\ \text{C's gain =}\frac{1}{5}\\ \text{A}=\frac{5}{10}-\frac{1}{10}=\frac{5-1}{10}=\frac{4}{10}\\ \text{B}=\frac{4}{10}-\frac{1}{10}=\frac{4-1}{10}=\frac{3}{10}\\ \text{C}=\frac{1}{10}+\frac{1}{5}=\frac{1+2}{10}=\frac{3}{10}\\ \text{A:B:C}=\text{4:3:3}\\ \mathbf{\text{Case 3}}\text{:}\\ \text{A:B:C}=\text{5:4:1}\left(\text{Old Ratio}\right)\\ \text{A:B:C}=\text{1:1:1}\left(\text{New Ratio}\right)\\ \text{A}=\frac{5}{10}-\frac{1}{3}=\frac{15-10}{30}=\frac{5}{30}\left(Sacrifice\right)\\ \text{B}=\frac{4}{10}-\frac{1}{3}=\frac{12-10}{30}=\frac{2}{30}\left(Sacrifice\right)\\ \text{C}=\frac{1}{10}-\frac{1}{3}=\frac{3-10}{30}=-\frac{7}{30}\left(Gain\right)\\ \mathbf{\text{Case 4}}\text{:}\\ \text{A:B:C}=\text{5:4:1}\left(\text{Old Ratio}\right)\\ \text{A's sacrifice to C}=\frac{5}{10}\times \frac{1}{10}=\frac{1}{20}\\ \text{B's sacrifice to C}=\frac{4}{10}\times \frac{1}{2}=\frac{4}{20}\\ \text{C's gain}=\frac{1}{20}+\frac{4}{20}=\frac{5}{20}\\ \text{A}=\frac{5}{10}-\frac{1}{20}=\frac{10-1}{20}=\frac{9}{20}\\ \text{B}=\frac{4}{10}-\frac{4}{20}=\frac{8-4}{20}=\frac{4}{20}\\ \text{C}=\frac{1}{10}+\frac{5}{20}=\frac{2+5}{20}=\frac{7}{20}\\ \text{A:B:C}=\text{9:4:7}\end{array}$