Old ratio of Praveena and Dhanya = 7:3 that is \(\frac{7}{10}:\frac{3}{10}\)
New ratio of Paveena, Dhanya and Malini = 5:2:3 that is, \(\frac{5}{10};\frac{2}{10};\frac{3}{10}\)
Share sacrificed = Old share - New share
Praveena \(=\frac { 7 }{ 10 } -\frac { 5 }{ 10 } =\frac { 7-5 }{ 10 } =\frac { 2 }{ 10 } \)
Dhanya \(=\frac { 3 }{ 10 } -\frac { 2 }{ 10 } =\frac { 3-2 }{ 10 } =\frac { 1 }{ 10 } \)
Sacrificing ratio ofPraveena and Dhanya is \(\frac { 2 }{ 10 } :\frac { 1 }{ 10 } \) that is 2:1

Computation of sacrificing ratio and new profit sharing ratio
Old ratio of Ambika, Dharani and Padma \(5:3:2\quad or\quad \frac { 5 }{ 10 } :\frac { 3 }{ 10 } :\frac { 2 }{ 10 } \)
Ramya's share of profit = 25% or \(\frac { 25 }{ 100 } or\frac { 1 }{ 4 } \)
Let the total share be 1 
Remaining share \(=1-\frac { 1 }{ 4 } =\frac { 4-1 }{ 4 } =\frac { 3 }{ 4 } \)
New share of old partners = Remaining share \(\times\) Old share
Ambika \(=\frac { 3 }{ 4 } \times \frac { 5 }{ 10 } =\frac { 15 }{ 40 } \)
Dharani \(=\frac { 3 }{ 4 } \times \frac { 3 }{ 10 } =\frac { 9 }{ 40 } \)
Padma \(=\frac { 3 }{ 4 } \times \frac { 2 }{ 10 } =\frac { 6 }{ 40 } \)
Share of new partner
Ramya \(=\frac { 1 }{ 4 } \)
In order to equalise the denominator, multiply and divide Ramya's share by 10 
Rarnyas share \(=\frac { 1 }{ 4 } \times \frac { 10 }{ 10 } =\frac { 10 }{ 40 } \)
New profit sharing ratio of Ambika, Dharani, Padma and Ramya \(=\frac { 15 }{ 40 } :\frac { 9 }{ 40 } :\frac { 6 }{ 40 } :\frac { 10 }{ 40 } \), that is 15: 9: 6: 10

Sacrificing ratio is the proportion of the profit which is sacrificed or foregone by the old partners in favour of the new partner. The purpose of finding the sacrificing ratio is to share the goodwill brought in by the new partner. 
Share sacrificed = Old share - New share
Sacrificing ratio = Ratio of share sacrificed by the old partners

As the sacrifice made by the existing partners is not mentioned, it is assumed that they sacrifice in their old profit sharing ratio 5:3. Therefore, sacrificing ratio is 5:3.

Computation of sacrificing ratio and new profit sharing ratio
Old ratio of Anil, Sunil and Hari = 4 : 3 : 3 or \(\frac { 4 }{ 10 } :\frac { 3 }{ 10 } :\frac { 3 }{ 10 } \)
Raja’s share of profit = 20% or 20/100 or 1/5
Let the total share be 1
Remaining share =  \(1-\frac { 1 }{ 5 } =\frac { 5-1 }{ 5 } \)
= \(\frac { 4 }{ 5 } \)
New share of old partners = Remaining share × Old share
Anil = \(\frac { 4 }{ 5 } \times \frac { 1 }{ 10 } =\frac { 16 }{ 50 } \)
Sunil = \(\frac { 4 }{ 5 } \times \frac { 3 }{ 10 } =\frac { 12 }{ 50 } \)
Hari =  \(\frac { 4 }{ 5 } \times \frac { 3 }{ 10 } =\frac { 12 }{ 50 } \)
Share of new partner Raja =  \(\frac { 1 }{ 5 } \)
In order to equalise the denominator, multiply and divide Raja’s share by 10
Raja’s share = \(\frac { 1 }{ 5 } \times \frac { 10 }{ 10 } =\frac { 10 }{ 50 } \)
New profit sharing ratio of Anil, Sunil, Hari and Raja = \(\frac { 16 }{ 50 } :\frac { 12 }{ 50 } :\frac { 12 }{ 50 } :\frac { 12 }{ 50 } \) that is, 8 : 6 : 6 : 5.

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, Suresh \(\frac { 3 }{ 5 } \) and Dinesh \(\frac{2}{5}\) \(Suresh\)
Share sacrificed = Old share x Proportion of share sacrificed
Suresh = \(\frac { 3 }{ 5 } \times \frac { 1 }{ 5 } =\frac { 3 }{ 25 } \)
Dinesh = \(\frac { 2 }{ 5 } \times \frac { 2 }{ 5 } =\frac { 4 }{ 25 } \)
Sacrificing ratio of Suresh and Dinesh is \(\frac { 3 }{ 25 } \) and \(\frac { 4 }{ 25 } \) that is, 3:4
New share = Old share - Share sacrificed
Suresh = \(\frac { 3 }{ 5 } -\frac { 3 }{ 25 } =\frac { 15-3 }{ 25 } =\frac { 12 }{ 25 } \)
Dinesh = \(\frac { 2 }{ 5 } -\frac { 4 }{ 25 } =\frac { 10-4 }{ 25 } =\frac { 6 }{ 25 } \)
Share of new partner = Sum of shares sacrificed by Suresh and Dinesh
Ramesh = \(\frac { 3 }{ 25 } +\frac { 4 }{ 25 } =\frac { 3+4 }{ 25 } =\frac { 7 }{ 25 } \)
New profit sharing ratio of Suresh, Dinesh and Ramesh = \(\frac { 12 }{ 25 } :\frac { 6 }{ 25 } :\frac { 7 }{ 25 } \)that is, 12:6:7.

Old ratio of Anbu and Raju = 3:2 that is, \(\frac { 3 }{ 5 } :\frac { 2 }{ 5 } \)
New ratio of Anbu, Raju and Akshai = 5:3:2, that is, \(\frac { 5 }{ 10 } :\frac { 3 }{ 10 } :\frac { 2 }{ 10 } \)
Share sacrificed = Old share - New share
Anbu = \(\frac { 3 }{ 5 } -\frac { 5 }{ 10 } =\frac { 6-5 }{ 10 } =\frac { 1 }{ 10 } \)
Raju =  \(\frac { 2 }{ 5 } -\frac { 3 }{ 10 } =\frac { 4-3 }{ 10 } =\frac { 1 }{ 10 } \)
Sacrificing ratio of Anbu and Raju is \(\frac { 1 }{ 10 } :\frac { 1 }{ 10 } \) that is 1:1

Ierald's share of goodwill = 45,000\(\times \frac{1}{3}\)
Rs. 15,000
As the sacrifice made by the existing partners is not mentioned, it is assumed that they sacrifice in their old profit sharing ratio is 1 : 1 : 1

According treatment for goodwill on admission of a partner is discussed below:
i) When new partner brings cash onwards goodwill
When the new partner brings cash towards goodwill in addition to the amount of capital it is distributed to the existing partners in the sacrificing ratio.
(ii) When the new partner does not bring goodwill in cash or in kind
If the new partner does not bring goodwill in cash or in kind, his share of goodwill must be adjusted through the capital accounts of the partners.
(iii) When the new partner brings only a part of the goodwill in cash or in kind
Sometimes the new partner may bring only a part of the goodwill in cash or assets. In such a case, for the cash or the assets brought the respective account is debited and for the amount not brought in cash or kind, the new partner's capital account is debited
(iv) Existing goodwill
If goodwill already appears in the books of accounts at the time of admission if the partners decide, it can be written off by transferring it to the existing partner's capital account/current account in the old profit sharing ratio.

The following adjustment are necessary at the time of admission of a partner
(i) Distribution of accumulated profits, reserves and losses.
(ii) Revaluation of assets and liabilities
(iii) Determination of new profit sharing ratio and sacrificing ratio
(iv) Adjustment for goodwill
(v) Adjustment of capital on the basis of new profit sharing ratio (if so agreed).

Good will of the firm is = Rs. 75,000
Rajesh share of goodwill = 75,000 \(\times\) \(\frac{1}{5}\)
= Rs. 15,000
It is to be distributed to Anbu and Shankar in their old ratio of 3:2

Since the sacrificing ratio is not given and the new partner’s share is given, it is assumed that the old profit sharing ratio (3:2) is the sacrificing ratio and the new partner’s share of goodwill is distributed to the old partners accordingly.

(d) (A) is false, but (R) is true

(a) Both (A) and (R) are true and (R) is correct explanation of (A)