Question

$\mathrm{A}$ and $\mathrm{B}$ are partners sharing profits and losses in the ratio of $3:2$. They admit $\mathrm{C}$ into the partnership for one-fourth share of the profits while $\mathrm{A}$ and $\mathrm{B}$ as between themselves are sharing profits & losses equally. Find the new profit sharing ratio and sacrificing ratio.

Answer 1

Step $1:$ Finding the new profit sharing ratio:

As, $\mathrm{C}$ is admitted for $\frac{1}{4}$ th shares.

Let the combined share of $\mathrm{A}$, $\mathrm{B}$and $\mathrm{C}$ $=1$.

Therefore, $\mathrm{C}$'s share $=\frac{1}{4}\times 1=\frac{1}{4}$

Thus, combined shares of $\mathrm{A}$ and $\mathrm{B}$ after $\mathrm{C}$’s admission $\text{=1-C's}$ share

$=1-\frac{1}{4}=\frac{3}{4}$

Therefore, New shares of $\mathrm{A}$ and $\mathrm{B}$ is

$\text{A=}\frac{3}{4}\times \frac{1}{2}=\frac{3}{8}$

$\text{B=}\frac{3}{4}\times \frac{1}{2}=\frac{3}{8}$

Therefore,

$\begin{array}{rcl}\mathrm{A}:\mathrm{B}:\mathrm{C}& =& \frac{3}{8}\mathrm{:}\frac{3}{8}\mathrm{:}\frac{1}{4}\\ & & =\mathrm{3:3:2}(\mathrm{multiplying}\mathrm{each}\mathrm{ratio}\mathrm{by}8)\end{array}$

Hence, the new profit sharing ratio is $3:3:2$.

Step $2:$ Finding the sacrificing ratio:

Old ratio ( $\mathrm{A}$and $\mathrm{B}$) $=3:2$

Sacrificing ratio = Old ratio – New ratio

$\mathrm{A}$’s sacrifice $=\frac{3}{5}-\frac{3}{8}=\frac{24-15}{40}=\frac{9}{40}$

$\mathrm{B}$’s sacrifice $=\frac{2}{5}-\frac{3}{8}=\frac{16-15}{40}=\frac{1}{40}$

Therefore, sacrificing ratio of $\mathrm{A}$and $\mathrm{B}$ is $=\frac{9}{40}:\frac{1}{40}=9:1$

Hence, the new profit sharing ratio is $3:3:2$ and the sacrificing ratio of $\mathrm{A}$and $\mathrm{B}$ is $9:1$.