Step 1: Understanding the Concept:
When a new partner joins a firm, they must receive a share of the total profits.
This share is provided by the existing partners, which is known as a "sacrifice."
The arrangement regarding how the new partner gets their share is crucial.
In many cases, partners sacrifice in their old ratio, but the problem can specify a different method.
Here, the phrase "acquires entirely from A" indicates a specific sacrifice.
Only Partner A is surrendering a portion of his interest to C.
Partner B's share in the total profit pool remains unchanged in terms of its fractional value.
The objective is to compute the remaining interest of A and express all shares with a common denominator.
Key Formula or Approach:
The fundamental relationship used in partnership changes is:
\[ \text{New Share} = \text{Old Share} - \text{Sacrificing Share} \]
We also need to ensure that the sum of all new shares equals 1 (the whole firm).
Step 2: Detailed Explanation:
1. Determine Old Profit Shares:
The profit-sharing ratio between A and B is $3 : 2$.
The total parts in the old ratio are $3 + 2 = 5$.
A's old share = $\frac{3}{5}$
B's old share = $\frac{2}{5}$
2. Identify the Sacrifice made for C:
C is admitted for a $\frac{1}{5}$th share of the total profits.
The problem explicitly states that C "acquires entirely from A."
This means A's sacrifice = $\frac{1}{5}$
Since B does not give up any share, B's sacrifice = $0$
3. Calculate the New Shares for each partner:
Partner A:
A's New Share = A's Old Share $-$ A's Sacrifice
\[ \text{A's New Share} = \frac{3}{5} - \frac{1}{5} = \frac{2}{5} \]
Partner B:
Since B made no sacrifice, B's share remains the same as the old share.
\[ \text{B's New Share} = \frac{2}{5} \]
Partner C:
C's share is given directly in the problem.
\[ \text{C's New Share} = \frac{1}{5} \]
4. Express as a Ratio:
We now list the shares of A, B, and C:
\[ A : B : C = \frac{2}{5} : \frac{2}{5} : \frac{1}{5} \]
Since the denominators are already identical (all are 5), we can directly state the ratio.
The New Profit Sharing Ratio is $2 : 2 : 1$.
Step 3: Final Answer:
By following the specific instructions of the acquisition, we find that A's share reduces while B's remains the same.
The resulting distribution of $2/5$ for A, $2/5$ for B, and $1/5$ for C gives us the final ratio of $2 : 2 : 1$.
This ratio will now be used for future profit distributions and for adjusting capitals.