Sure, let's go through this step by step.
### Step 1: Initial Profit Sharing Ratio
Initially, partners A, B, and C share the profits in the ratio 3:2:1.
- Let the total profit be P, then A gets 3 parts, B gets 2 parts, and C gets 1 part.
- Total number of parts = 3 + 2 + 1 = 6 parts.
Therefore:
- A's share of profit: [tex]\(\frac{3}{6}\)[/tex] of P = 0.5P
- B's share of profit: [tex]\(\frac{2}{6}\)[/tex] of P = [tex]\(\frac{1}{3}\)[/tex]P
- C's share of profit: [tex]\(\frac{1}{6}\)[/tex] of P = [tex]\(\frac{1}{6}\)[/tex]P
### Step 2: Surrendered Shares
A surrenders [tex]\(\frac{1}{3}\)[/tex] of his share to C:
- A's surrendered share = [tex]\(\frac{1}{3} \times \frac{3}{6}P = \frac{1}{6}P\)[/tex]
B surrenders [tex]\(\frac{1}{4}\)[/tex] of his share to C:
- B's surrendered share = [tex]\(\frac{1}{4} \times \frac{2}{6}P = \frac{1}{12}P\)[/tex]
### Step 3: New Shares After Surrender
Now, let's calculate the new shares:
- A's new share = [tex]\(\frac{3}{6}P - \frac{1}{6}P = \frac{2}{6}P = \frac{1}{3}P\)[/tex]
- B's new share = [tex]\(\frac{2}{6}P - \frac{1}{12}P = \frac{4}{12}P - \frac{1}{12}P = \frac{3}{12}P = \frac{1}{4}P\)[/tex]
- C's new share = Original share + surrendered shares = [tex]\(\frac{1}{6}P + \frac{1}{6}P + \frac{1}{12}P = \frac{2}{6}P + \frac{1}{12}P = \frac{4}{12}P + \frac{1}{12}P = \frac{5}{12}P\)[/tex]
### Step 4: Calculate New Profit Sharing Ratio
Total new profit shares:
- A's share: [tex]\(\frac{1}{3}P\)[/tex]
- B's share: [tex]\(\frac{1}{4}P\)[/tex]
- C's share: [tex]\(\frac{5}{12}P\)[/tex]
To find a common ratio, we convert these to a common denominator:
- A's share as a ratio of the total: [tex]\(\frac{4}{12}P\)[/tex]
- B's share as a ratio of the total: [tex]\(\frac{3}{12}P\)[/tex]
- C's share as a ratio of the total: [tex]\(\frac{5}{12}P\)[/tex]
The simplified ratio is thus:
- A : B : C = 4 : 3 : 5
### Step 5: Calculate Sacrifice or Gain
- A's sacrifice = [tex]\(\frac{1}{6}P\)[/tex]
- B's sacrifice = [tex]\(\frac{1}{12}P\)[/tex]
- C's gain = A's surrender + B's surrender = [tex]\(\frac{1}{6}P + \frac{1}{12}P = \frac{2}{12}P + \frac{1}{12}P = \frac{3}{12}P\)[/tex]
### Final Answer
The new profit sharing ratio is 4:3:5.
- A's sacrifice: [tex]\(\frac{1}{6} = \frac{2}{12}\)[/tex]
- B's sacrifice: [tex]\(\frac{1}{12}\)[/tex]
- C's gain: [tex]\(\frac{3}{12}\)[/tex]
So the detailed step-by-step solution shows that the new profit sharing ratio is 4:3:5, with A's sacrifice being [tex]\(\frac{2}{12}\)[/tex], B's sacrifice being [tex]\(\frac{1}{12}\)[/tex], and C's gain being [tex]\(\frac{3}{12}\)[/tex].
