Computation of sacrificing ratio and new profit sharing ratio
Arun’s share = \(\frac { 1 }{ 4 } \)
Proportion of share sacrificed = 1:1(equally) i.e. \(\frac { 1 }{ 2 } :\frac { 1 }{ 2 } \)
Share sacrificed = New partner’s share × Proportion of share sacrificed
Mahesh = \(\frac { 1 }{ 4 } \times \frac { 1 }{ 2 } =\frac { 1 }{ 8 } \)
Dhanush = \(\frac { 1 }{ 4 } \times \frac { 1 }{ 2 } =\frac { 1 }{ 8 } \)
Sacrificing ratio of Mahesh and Dhanush is \(\frac { 1 }{ 8 } :\frac { 1 }{ 8 } \) that is, 1:1
New share of old partner = Old share - Share sacrificed
Mahesh = \(\frac { 2 }{ 3 } -\frac { 1 }{ 8 } =\frac { 16-3 }{ 24 } =\frac { 13 }{ 24 } \)
Dhanush = \(\frac { 1 }{ 3 } -\frac { 1 }{ 8 } =\frac { 8-3 }{ 24 } =\frac { 5 }{ 24 } \)
Share of new partner Arun = \(\frac { 1 }{ 4 } \)
In order to equate, multiply and divide Arun’s share by 6
= \(\frac { 1 }{ 4 } \times \frac { 6 }{ 6 } =\frac { 6 }{ 24 } \)
New profit sharing ratio of Mahesh, Dhanush and Arun = \(\frac { 13 }{ 24 } :\frac { 5 }{ 24 } :\frac { 6 }{ 24 } \)that is, 13:5:6.

Computation of sacrificing ratio and new profit sharing ratio
Old share = 3:2 that is, Prasanth \(\frac { 3 }{ 5 } \) and Nisha \(\frac { 2 }{ 5 } \)
Share sacrificed = Old share × Proportion of share sacrificed
Prasanth = \(\frac { 3 }{ 5 } \times \frac { 2 }{ 5 } =\frac { 6 }{ 25 } \)
Nisha = \(\frac { 2 }{ 5 } \times \frac { 2 }{ 5 } =\frac { 4 }{ 25 } \)
Sacrificing ratio of Prasanth and Nisha is \(\frac { 6 }{ 25 } \) and \(\frac { 4 }{ 25 } \), that is, 3:2
New share = Old share - Share sacrificed
Prasanth = \(\frac { 3 }{ 5 } -\frac { 6 }{ 25 } =\frac { 15-6 }{ 25 } =\frac { 9 }{ 25 } \)
Nisha = \(\frac { 2 }{ 5 } -\frac { 4 }{ 25 } =\frac { 10-4 }{ 25 } =\frac { 6 }{ 25 } \)
Share of new partner = Sum of shares sacrificed by Prasanth and Nisha
Ramya = \(\frac { 6 }{ 25 } +\frac { 4 }{ 25 } =\frac { 6+4 }{ 25 } =\frac { 10 }{ 25 } \)
New profit sharing ratio of Prasanth, Nisha and Ramya = \(\frac { 9 }{ 25 } :\frac { 6 }{ 25 } :\frac { 10 }{ 25 } \) that is, 9:6:10

Computation of sacrificing ratio and new profit sharing ratio
Share sacrificed =  \(\frac { 1 }{ 5 } ,\frac { 1 }{ 5 } \)
Sacrificing ratio of Hameed and Govind is 1:1
Old ratio is 5:3 that is \(\frac { 5 }{ 8 } :\frac { 3 }{ 8 } \)
New share of old partner = Old share - Share sacrificed
Hameed = \(\frac { 5 }{ 8 } -\frac { 1 }{ 5 } =\frac { 25-8 }{ 40 } =\frac { 17 }{ 40 } \)
Govind = \(\frac { 3 }{ 8 } -\frac { 1 }{ 5 } =\frac { 15-8 }{ 40 } =\frac { 7 }{ 40 } \)
Share of new partner
John = Sum of shares sacrificed by old partners
=\(\frac { 1 }{ 5 } +\frac { 1 }{ 5 } =\frac { 2 }{ 5 } \)
In order to equalise the denominator of John’s share, multiply and divide by 8 John’s share =\(\\ \frac { 2 }{ 5 } \times \frac { 8 }{ 8 } =\frac { 16 }{ 40 } \)
New profit sharing ratio of Hameed, Govind and John is \(\frac { 17 }{ 40 } :\frac { 7 }{ 40 } :\frac { 16 }{ 40 } \) or 17:7:16

Computation of sacrificing ratio and new profit sharing ratio
Share sacrificed by old partners
Hari =\(\frac { 1 }{ 8 } \)
Saleem = 0
Sacrificing ratio = 1:0
Old ratio of Hari and Saleem is 5:3 that is \(\frac { 5 }{ 8 } :\frac { 3 }{ 8 } \)
New share of old partner = Old share - Share sacrificed
Hari = \(\frac { 5 }{ 8 } -\frac { 1 }{ 8 } =\frac { 5-1 }{ 8 } =\frac { 4 }{ 8 } \)
Saleem = \(\frac { 3 }{ 8 } \)
Share of new partner Joel = \(\frac { 1 }{ 8 } \)
New profit sharing ratio of Hari, Saleem and Joel is \(\frac { 4 }{ 8 } :\frac { 3 }{ 8 } :\frac { 1 }{ 8 } \) that is, 4:3:1.