Calculating the Sacrificing Ratio in Partnership Admission

When a new partner is admitted into a firm, the existing partners usually have to give up or 'sacrifice' a portion of their share of profits to the new partner. The ratio in which the old partners sacrifice their share of profit is known as the sacrificing ratio. This ratio is often used to distribute the goodwill brought in by the new partner.

Understanding the Problem

We are given:

• Old partners: Anita and Bindu.
• Old profit-sharing ratio: 3:2.
• New partner: Meria, admitted for 1/4th share.
• New profit-sharing ratio between Anita and Bindu is 2:1. This means the remaining profit after giving Meria her share will be divided between Anita and Bindu in the ratio 2:1.

We need to calculate the sacrificing ratio of Anita and Bindu.

Step-by-Step Calculation

Step 1: Determine the total share of profit for the old partners after the new partner's admission.

The total profit of the firm is considered as 1. The new partner, Meria, gets 1/4th share. The remaining share belongs to the old partners, Anita and Bindu.

Total share = 1

Meria's share = \( \frac{1}{4} \)

Remaining share for Anita and Bindu = \( 1 - \text{Meria's share} = 1 - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4} \)

So, Anita and Bindu will share the remaining 3/4th of the profit.

Step 2: Calculate the new profit share of each old partner.

Anita and Bindu share the remaining 3/4th profit in their new ratio of 2:1. The total of their new ratio is \(2 + 1 = 3\).

Anita's new share = Remaining share \( \times \) Anita's new ratio in remaining share

\( = \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \)

Bindu's new share = Remaining share \( \times \) Bindu's new ratio in remaining share

\( = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \)

The new profit-sharing ratio for all partners (Anita: Bindu: Meria) is \( \frac{1}{2} : \frac{1}{4} : \frac{1}{4} \). To get a common denominator (4), this is \( \frac{2}{4} : \frac{1}{4} : \frac{1}{4} \), or 2:1:1.

Step 3: Calculate the sacrificing share of each old partner.

Sacrificing share = Old share - New share

First, find the old shares:

Old ratio of Anita and Bindu = 3:2. Total of old ratio = \(3 + 2 = 5\).

Anita's old share = \( \frac{3}{5} \)

Bindu's old share = \( \frac{2}{5} \)

Now, calculate the sacrifice for each partner:

Anita's sacrifice = Old share - New share

\( = \frac{3}{5} - \frac{1}{2} \)

To subtract fractions, find the least common multiple (LCM) of the denominators (5 and 2), which is 10.

\( = \frac{3 \times 2}{5 \times 2} - \frac{1 \times 5}{2 \times 5} = \frac{6}{10} - \frac{5}{10} = \frac{6-5}{10} = \frac{1}{10} \)

Bindu's sacrifice = Old share - New share

\( = \frac{2}{5} - \frac{1}{4} \)

To subtract fractions, find the LCM of the denominators (5 and 4), which is 20.

\( = \frac{2 \times 4}{5 \times 4} - \frac{1 \times 5}{4 \times 5} = \frac{8}{20} - \frac{5}{20} = \frac{8-5}{20} = \frac{3}{20} \)

Step 4: Determine the sacrificing ratio.

The sacrificing ratio is the ratio of the sacrifices made by Anita and Bindu.

Sacrificing Ratio (Anita : Bindu) = Anita's sacrifice : Bindu's sacrifice

\( = \frac{1}{10} : \frac{3}{20} \)

To express this ratio in whole numbers, find a common denominator or multiply both sides by the LCM of the denominators (10 and 20), which is 20.

\( (\frac{1}{10} \times 20) : (\frac{3}{20} \times 20) \)

\( = 2 : 3 \)

The sacrificing ratio of Anita and Bindu is 2:3.

Summary of Shares

Sacrificing Ratio (Anita : Bindu) = \( \frac{1}{10} : \frac{3}{20} = 2:3 \)

Revision Table: Partnership Sacrificing Ratio

Additional Information: Impact of New Partner Admission

The admission of a new partner leads to several adjustments in the partnership accounts:

• Change in Profit Sharing Ratio: The old ratio changes to a new ratio among all partners, including the new one.
• Sacrificing Ratio and Gaining Ratio: Old partners typically sacrifice a share of profit, leading to a sacrificing ratio. In some specific cases, an old partner might gain, resulting in a gaining ratio. The new partner gains a share of profit.
• Accounting for Goodwill: Goodwill is adjusted through the sacrificing ratio to compensate the sacrificing partners for their loss of profit share.
• Revaluation of Assets and Liabilities: Assets and liabilities are revalued to reflect their current values, and the profit or loss on revaluation is distributed among old partners in their old profit-sharing ratio.
• Adjustment of Accumulated Profits and Reserves: Undistributed profits and reserves are distributed among old partners in their old profit-sharing ratio.
• Adjustment of Capital: Capital accounts are adjusted based on the new profit-sharing ratio or other agreed terms.