**Solution**

**(i)**

| R | T |

OLD RATIO | 3 | 2 |

**Sacrificing Ratio = Old Ratio × Surrender Ratio**

R’s = 3/5 × 1/4 = 3/20

T’s = 2/5 × 1/5 = 2/25

**New Ratio = Old Ratio − Sacrificing Ratio**

R = 3/5-3/20 = 9/20

T = 2/5-2/25 = 8/25

S’s Share = R’s Sacrifice + S’s Sacrifice

= 3/20 + 2/25 = 23/100

R | T | S | |

New profit-sharing ratio | 9/20 | 8/25 | 23/100 |

= | 45/100 | 32/100 | 23/100 |

= | 45 | 32 | 23 |

**(ii)**

| A | B |

OLD RATIO | 1 | 1 |

C admits for 1/4^{th} share of profit

Let the combined share of A, B and C be = 1

Combined share of A and B = 1-C’s share = 1-1/4 = 3/4

**New Ratio = Combined share of A and B ×2/3**

A’s = 3/4 × 2/3 = 6/12

B’s = 3/4 × 1/3 = 3/12

A | B | C | |

New profit-sharing ratio | 6/12 | 3/12 | 1/4 |

= | 6/12 | 3/12 | 3/12 |

= | 2 | 1 | 1 |

(iii)

| A | B |

OLD RATIO | 3 | 2 |

** C admits for 1/5share of profit**

A’s sacrifice | = C’s share × 1/5 |

= 1/5 × 1/5 | |

= 1/25 | |

B’s sacrifice | = C’s share × 4/5 |

= 1/5 × 4/5 | |

= 4/25 |

**New Ratio = Old Ratio − Sacrificing Ratio**

A’s = 3/5-1/25 = 14/25

B’s = 2/5-4/25 = 6/25

A | B | C | |

New profit-sharing ratio | 14/25 | 6/25 | 1/5 |

= | 14/25 | 6/25 | 5/25 |

= | 14 | 6 | 5 |

**(iv)**

| X | Y | Z |

OLD RATIO | 3 | 2 | 1 |

W admits for 1/6 share of profit

Let combined share of all partner after W’s admission be = 1

Combined share X and Y in the new firm = 1- Z’s share- W’s share

= 1-1/6-1/6 = 4/6

**New Ratio = Old Ratio × Combined share of X and Y**

A’s | = 3/5×4/6 |

= 12/30 | |

B’s | = 2/5×4/6 |

= 8/30 |

X | Y | Z | W | |

New profit-sharing ratio | 12/30 | 8/30 | 1/6 | 1/6 |

= | 12/30 | 8/30 | 5/30 | 5/30 |

= | 12 | 8 | 5 | 5 |

**(v)**

| A | B |

OLD RATIO | 1 | 1 |

C admits for 1/5 share

D admits for 1/6 share

Let combined share of all partner after C and D’s admission be = 1

Combined share of profit of A and B after C and D’s admission = 1- C’s share – Z’s share

= 1-1/5-1/6 = 19/30

**New Ratio = Old Ratio × Combined share of A and B**

A’s | = 1/2 × 19/30 |

= 19/60 | |

B’s | = 1/2 × 19/30 |

= 19/60 |

X | Y | Z | W | |

New profit-sharing ratio | 19/60 | 19/60 | 1/5 | 1/6 |

= | 19/60 | 19/60 | 12/60 | 10/60 |

= | 19 | 19 | 12 | 10 |

**(vi)**

| A | B |

OLD RATIO | 3 | 2 |

C admits for 1/5 share of profit

Let the combined share of all partners after C’s admission be = 1

Combined share of A and B after C’s admission = 1 – C’s share

= 1-1/4 = 3/4

**New ratio of A and B each= Combined share of A and B × 1/2**= 3/4 × 1/2 = 3/8 each

X | Y | Z | |

New profit-sharing ratio | 3/8 | 3/8 | 1/4 |

= | 3/8 | 3/8 | 2/8 |

= | 3 | 3 | 2 |