A, B , and C are partners sharing profits and losses in the ratio of \[5:4:1\] . Calculate new profit sharing ratio, sacrificing ratio in the following case:
Case: \[A\] , \[B\] , and \[C\] will share future profits and losses equally.
A. New profit sharing ratio – $ 3:4:3 $ , sacrifice of \[A - \dfrac{1}{5}\] ; Gain of \[C - \dfrac{1}{5}\] ;
B. New profit sharing ratio – $ 4:3:3 $ ; sacrificing ratio ( \[A\] and \[B\] ) $ - 1:1 $ ; Gain of \[\left( { - \dfrac{2}{{12}}} \right)\] ;
C. New profit sharing ratio – Equal; sacrificing ratio ( \[A\] and \[B\] ) $ - 5:2 $ ; Gain of \[C - \dfrac{7}{{30}}\] ;
D. New profit sharing ratio - $ 9:4:7 $ ; sacrificing ratio ( \[A\] and \[B\] ) $ - 1:4 $ ; Gain of \[C - \dfrac{5}{{22}}\] ;

Hint: To solve the given question comparing the ratio of themselves for that the given problems can be solved by arising cases like a case $ I $ , case $ II $ , case $ III $ , and case $ IV $ . Four cases are arising in that problem. We can be comparing the ratio of their sharing and gaining. In that way, a particular solution can find out. Let us discuss the cases one by one.

Complete step by step solution:
The given problem is based on profit and loss of sharing and gaining. The problem can be solved by using their ratios...
Case 1: Consider the ratio of \[A\] , \[B\] , and \[C\] as \[A:B:C\] will be equal \[5:4:1\]
as \[C\] acquires $ \dfrac{1}{5}th $ from \[A\]
 \[A\] can sacrifice of 1th part and \[C\] can given 1th part so, solving their fractions we get
 \[ \Rightarrow \,\,A = \dfrac{5}{{10}} - \dfrac{1}{5}\]
taking L.C.M of denominator 5,10 we get
 $ \Rightarrow \,\,A = \dfrac{{5 - 2}}{{10}} $
 $ \Rightarrow \,\,A = \dfrac{3}{{10}} $
as the denominator will be the sum of ratios $ 5 + 4:1 $ as $ 5 + 4 + 1 $ equal to 10.
as \[B\] will be equal to \[\dfrac{4}{{10}}\]
as \[C\] will be equal to $ \dfrac{1}{{10}} + \dfrac{1}{5} = \dfrac{3}{{10}} $ \[\]
as the ratio of \[A\] , \[B\] , and \[C\] will be \[A:B:C\] are $ 3:4:3 $ ,
Case 2: In the second case the ratio of \[A\] , \[B\] , and \[C\] are \[5:4:1\] .
Let \[C\] acquires $ \dfrac{1}{5}th $ share equally from \[A\]
 \[A\] sacrifice the share will be $ \dfrac{1}{{10}} $
 \[B\] sacrifice the share will be $ \dfrac{1}{{10}} $
 \[C\] gaining we get $ \dfrac{1}{5} $
Solving \[A\] , \[B\] , and \[C\] we get the values of ratios
 \[ \Rightarrow \,\,A = \dfrac{5}{{10}} - \dfrac{1}{{10}}\] as taking L.C.M. of denominator
 $ \Rightarrow \,\,A = \dfrac{4}{{10}} $
As \[B\] will be $ \dfrac{4}{{10}} - \dfrac{1}{{10}} $
 $ \Rightarrow \,\,\dfrac{{4 - 1}}{{10}} $
 $ \Rightarrow \,\,\dfrac{3}{{10}} $
As \[C\] will be equal to $ \dfrac{1}{{10}} + \dfrac{1}{5} $
 $ \Rightarrow \,\,\dfrac{{1 + 2}}{{10}} $
 $ \Rightarrow \,\,\dfrac{3}{{10}} $
The ratio of \[A\] , \[B\] , and \[C\] are \[4:3:3\]
 Case 3: In the third case considered both the old ratio and new ratio of \[A\] , \[B\] , and \[C\] we get
The old ratio of \[A\] , \[B\] , and \[C\] are \[5:4:1\] \[\]
The new ratio of \[A\] , \[B\] , and \[C\] are \[1:1:1\]
The value of \[A\] sacrifice will be \[\dfrac{5}{{10}} - \dfrac{1}{3}\]
as denominator is 10,3 taking L.C.M of them = \[\dfrac{{15 - 10}}{{30}}\]
The value of \[B\] sacrifice will be.
 \[ \Rightarrow \,\,\dfrac{4}{{10}} - \dfrac{1}{3}\]
Taking L.C.M of denominator of above term we get
 \[ \Rightarrow \,\,\dfrac{{12 - 10}}{{30}}\]
 \[ \Rightarrow \,\,\dfrac{2}{{30}}\]
The value of \[C\] gaining will be
 \[ \Rightarrow \,\,\dfrac{1}{{10}} - \dfrac{1}{3}\]
 \[ \Rightarrow \,\,\dfrac{{3 - 10}}{{30}}\]
 \[ \Rightarrow \,\,\dfrac{{ - 7}}{{30}}\]
Case 4: In the fourth case taking the old ratio of \[A\] , \[B\] , and \[C\] are \[5:4:1\]
as \[A\] sacrifices to \[C\] we get
 \[ \Rightarrow \,\,\dfrac{5}{{10}} \times \dfrac{1}{{10}}\]
 \[ \Rightarrow \,\,\dfrac{5}{{100}}\]
 \[ \Rightarrow \,\,\dfrac{1}{{20}}\]
as \[B\] sacrifices to \[C\] we get
 \[ \Rightarrow \,\,\dfrac{4}{{10}} \times \dfrac{1}{2}\]
 \[ \Rightarrow \,\,\dfrac{4}{{20}}\]
as \[C\] gains we get \[\dfrac{1}{{20}} + \dfrac{4}{{20}} = \dfrac{5}{{20}}\]
as the values of \[A\] , \[B\] , and \[C\] for solving we get
 \[ \Rightarrow \,\,A = \dfrac{5}{{10}} - \dfrac{1}{{20}} = \dfrac{{10 - 1}}{{20}} = \dfrac{9}{{20}}\]
 \[ \Rightarrow \,\,B = \dfrac{4}{{10}} - \dfrac{4}{{20}} = \dfrac{{8 - 4}}{{20}} = \dfrac{4}{{20}}\]
 \[ \Rightarrow \,\,C = \dfrac{1}{{10}} + \dfrac{5}{{10}} = \dfrac{{2 + 5}}{{10}} = \dfrac{7}{{20}}\]
The ratio of \[A:B:C\] are \[9:4:7\]

Note: In the given problem profit, a loss is in the ratio \[5:4:1\] for solving the old ratio. Then four cases arise having old ratios and also a new ratio. By solving that we can find out the general solution of the given problem as sacrificing ratio = old ratio = new ratio and identity of gaining ratio = new ratio – old ratio. In that way, the solution of the given problem can be finding out