Answer :
Let's solve this problem step-by-step.
### Step-by-Step Solution
Initially, [tex]$A$[/tex] and [tex]$B$[/tex] share profit and loss equally. This means their initial ratios are:
- [tex]\( \text{Initial Ratio of } A = \frac{1}{2} \)[/tex]
- [tex]\( \text{Initial Ratio of } B = \frac{1}{2} \)[/tex]
### Step 1: C's Admission Share
[tex]$C$[/tex] is admitted for a [tex]\(\frac{1}{6}\)[/tex]th share in the profits and losses.
So, [tex]\( \text{C's Share} = \frac{1}{6} \)[/tex].
### Step 2: Grimm Share by B
Half of C's share is given by [tex]$B$[/tex]:
[tex]\[ \text{Share of C grimmed by B} = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \][/tex]
### Step 3: Remaining Share of C
The remaining share of C is:
[tex]\[ \text{Remaining Share of C} = \frac{1}{6} - \frac{1}{12} = \frac{1}{12} \][/tex]
### Step 4: Contribution to Remaining Share
The remaining share of C is contributed by [tex]$A$[/tex] and [tex]$B$[/tex] where the contribution of [tex]$B$[/tex] is thrice that of [tex]$A$[/tex].
Let the contribution of [tex]$A$[/tex] towards the remaining share be [tex]\(x\)[/tex]. Therefore, the contribution of [tex]$B$[/tex] will be [tex]\(3x\)[/tex].
[tex]\[ x + 3x = \frac{1}{12} \][/tex]
### Step 5: Solving for x
[tex]\[ 4x = \frac{1}{12} \][/tex]
[tex]\[ x = \frac{1}{12} \div 4 = \frac{1}{12} \times \frac{1}{4} = \frac{1}{48} \][/tex]
So, the contribution made by [tex]$A$[/tex] is [tex]\(\frac{1}{48}\)[/tex] and the contribution made by [tex]$B$[/tex] is [tex]\(3 \times \frac{1}{48} = \frac{3}{48} = \frac{1}{16}\)[/tex].
### Final Ratios after Admission of C
Now we calculate the final ratios for each partner after C is admitted.
### Step 6: Calculating Final Ratios
#### For A:
[tex]\[ \text{Initial Ratio of A} = \frac{1}{2} \][/tex]
[tex]\[ \text{Contribution of A} = \frac{1}{48} \][/tex]
[tex]\[ \text{Final Ratio of A} = \frac{1}{2} - \frac{1}{48} \][/tex]
[tex]\[ \text{Final Ratio of A} = \frac{24}{48} - \frac{1}{48} = \frac{23}{48} \][/tex]
#### For B:
[tex]\[ \text{Initial Ratio of B} = \frac{1}{2} \][/tex]
[tex]\[ \text{Share grimmed by B} = \frac{1}{12} \][/tex]
[tex]\[ \text{Contribution by B} = \frac{3}{48} = \frac{1}{16} \][/tex]
[tex]\[ \text{Final Ratio of B} = \frac{1}{2} - \frac{1}{12} - \frac{1}{16} \][/tex]
[tex]\[ \text{Final Ratio of B} = \frac{24}{48} - \frac{4}{48} - \frac{3}{48} = \frac{24 - 4 - 3}{48} = \frac{17}{48} \][/tex]
#### For C:
[tex]\[ \text{Final Ratio of C} = \frac{1}{6} = \frac{8}{48} \][/tex]
### Summary of Final Ratios
- [tex]\( \text{Final Ratio of A} = \frac{23}{48} \approx 0.4791666666666667 \)[/tex]
- [tex]\( \text{Final Ratio of B} = \frac{17}{48} \approx 0.3541666666666667 \)[/tex]
- [tex]\( \text{Final Ratio of C} = \frac{8}{48} = \frac{1}{6} \approx 0.16666666666666666 \)[/tex]
Thus, after admitting C, the final profit-sharing ratios are:
- A: [tex]\( \approx 0.4791666666666667\)[/tex]
- B: [tex]\( \approx 0.3541666666666667\)[/tex]
- C: [tex]\( \approx 0.16666666666666666\)[/tex]
### Step-by-Step Solution
Initially, [tex]$A$[/tex] and [tex]$B$[/tex] share profit and loss equally. This means their initial ratios are:
- [tex]\( \text{Initial Ratio of } A = \frac{1}{2} \)[/tex]
- [tex]\( \text{Initial Ratio of } B = \frac{1}{2} \)[/tex]
### Step 1: C's Admission Share
[tex]$C$[/tex] is admitted for a [tex]\(\frac{1}{6}\)[/tex]th share in the profits and losses.
So, [tex]\( \text{C's Share} = \frac{1}{6} \)[/tex].
### Step 2: Grimm Share by B
Half of C's share is given by [tex]$B$[/tex]:
[tex]\[ \text{Share of C grimmed by B} = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \][/tex]
### Step 3: Remaining Share of C
The remaining share of C is:
[tex]\[ \text{Remaining Share of C} = \frac{1}{6} - \frac{1}{12} = \frac{1}{12} \][/tex]
### Step 4: Contribution to Remaining Share
The remaining share of C is contributed by [tex]$A$[/tex] and [tex]$B$[/tex] where the contribution of [tex]$B$[/tex] is thrice that of [tex]$A$[/tex].
Let the contribution of [tex]$A$[/tex] towards the remaining share be [tex]\(x\)[/tex]. Therefore, the contribution of [tex]$B$[/tex] will be [tex]\(3x\)[/tex].
[tex]\[ x + 3x = \frac{1}{12} \][/tex]
### Step 5: Solving for x
[tex]\[ 4x = \frac{1}{12} \][/tex]
[tex]\[ x = \frac{1}{12} \div 4 = \frac{1}{12} \times \frac{1}{4} = \frac{1}{48} \][/tex]
So, the contribution made by [tex]$A$[/tex] is [tex]\(\frac{1}{48}\)[/tex] and the contribution made by [tex]$B$[/tex] is [tex]\(3 \times \frac{1}{48} = \frac{3}{48} = \frac{1}{16}\)[/tex].
### Final Ratios after Admission of C
Now we calculate the final ratios for each partner after C is admitted.
### Step 6: Calculating Final Ratios
#### For A:
[tex]\[ \text{Initial Ratio of A} = \frac{1}{2} \][/tex]
[tex]\[ \text{Contribution of A} = \frac{1}{48} \][/tex]
[tex]\[ \text{Final Ratio of A} = \frac{1}{2} - \frac{1}{48} \][/tex]
[tex]\[ \text{Final Ratio of A} = \frac{24}{48} - \frac{1}{48} = \frac{23}{48} \][/tex]
#### For B:
[tex]\[ \text{Initial Ratio of B} = \frac{1}{2} \][/tex]
[tex]\[ \text{Share grimmed by B} = \frac{1}{12} \][/tex]
[tex]\[ \text{Contribution by B} = \frac{3}{48} = \frac{1}{16} \][/tex]
[tex]\[ \text{Final Ratio of B} = \frac{1}{2} - \frac{1}{12} - \frac{1}{16} \][/tex]
[tex]\[ \text{Final Ratio of B} = \frac{24}{48} - \frac{4}{48} - \frac{3}{48} = \frac{24 - 4 - 3}{48} = \frac{17}{48} \][/tex]
#### For C:
[tex]\[ \text{Final Ratio of C} = \frac{1}{6} = \frac{8}{48} \][/tex]
### Summary of Final Ratios
- [tex]\( \text{Final Ratio of A} = \frac{23}{48} \approx 0.4791666666666667 \)[/tex]
- [tex]\( \text{Final Ratio of B} = \frac{17}{48} \approx 0.3541666666666667 \)[/tex]
- [tex]\( \text{Final Ratio of C} = \frac{8}{48} = \frac{1}{6} \approx 0.16666666666666666 \)[/tex]
Thus, after admitting C, the final profit-sharing ratios are:
- A: [tex]\( \approx 0.4791666666666667\)[/tex]
- B: [tex]\( \approx 0.3541666666666667\)[/tex]
- C: [tex]\( \approx 0.16666666666666666\)[/tex]
