Question

$A$ and $B$ are partners sharing profit and losses in the ratio $3:2$. They decided to admit $C$ as a partner for $\dfrac{1}{4}th$ share . New ratio and sacrificing ratio will be
A. $9:7:6,\,3:2$
B. $3:2:1,\,3:2$
C. $10:5:5,\,3:2$
D. $3:3:2,9:1$

Answer 1

Hint:We know that a partnership is a formula agreement by two or more parties to manage and operate a business and share its profit. In general partnership, each partner shares the profits and risks of the loss. We can solve the question with the given ratio. The formula for Sacrificing Ratio is Old ratio$ - $New ratio.

Complete step by step answer:
We need to first understand the requirement of the question which is the sacrificing and new ratio. We are given that the old ratio of $A$ and $B$ is $3:2$ and C is admitted for the $\dfrac{1}{4}th$ of the share. Let us assume that the combined share of A,B and C is $1$.

So the combined share of $A$ and $B$ after $C’s$ admission is $1 - C's$ share.
By putting the value , we get: $1 - \dfrac{1}{4} = \dfrac{{4 - 1}}{4} = \dfrac{3}{4}$.
This will change the ratio of the previous partners.
So the new share of A $ = \dfrac{3}{4} \times \dfrac{1}{2} = \dfrac{3}{8}$,
and the new share of B $ = \dfrac{3}{4} \times \dfrac{1}{2} = \dfrac{3}{8}$.
Since $C$ is admitted here, his new share is $\dfrac{1}{4}$.
Therefore the new ratio is $A:B:C = \dfrac{3}{8}:\dfrac{3}{8}:\dfrac{1}{4}$.
By creating the denominator of all three shares as same:
$\dfrac{3}{8}:\dfrac{3}{8}:\dfrac{{1 \times 2}}{{4 \times 2}} = \dfrac{3}{8}:\dfrac{3}{8}:\dfrac{2}{8}$

So the new ratio of $A, B$ and $C$ is $3:3:2$.
Now the formula says that Sacrificing Ratio is an Old ratio$ - $New ratio.
The old ratio of $A$ and $B$ is $3:2$.
So the old ratio of $A$ is $\dfrac{3}{{3 + 2}} = \dfrac{3}{5}$
and the old ratio of $B$ is $\dfrac{2}{{3 + 2}} = \dfrac{2}{5}$.
By substituting the values we have sacrificing ratio of $A$ is,
$\dfrac{3}{5} - \dfrac{3}{8} = \dfrac{{24 - 15}}{{40}} = \dfrac{9}{{40}}$
Similarly the sacrificing ratio of $B$ is,
$\dfrac{2}{5} - \dfrac{3}{8} = \dfrac{{16 - 15}}{{40}} = \dfrac{1}{{40}}$
Therefore the sacrificing ratio of $A$ and $B$ together is $\dfrac{9}{{40}}:\dfrac{1}{{40}} = 9:1$.

Hence the new ratio is $3:3:2$ and the sacrificing ratio is $9:1$ and the correct option is D.

Note: We should note that there is no sacrificing ratio for $C$ as he did not sacrifice his shares. On his admission, the old partners $A$ and $B$ sacrificed their old ratio. We should keep in mind that the new partner purchases his profit from the old partners, so the new profit sharing ratio can be calculated by deducting the sacrifice made by the existing partners.