Calculating New Profit Sharing Ratio on Partner Admission

When a new partner is admitted into a firm, the existing partners usually give up a portion of their share of profit to the new partner. This changes the profit-sharing ratio among all partners, including the new one. The amount given up by the old partners is called their sacrifice, and the ratio in which they sacrifice is called the sacrifice ratio.

In this problem, we are given the old profit-sharing ratio of partners A, B, and C, the share of the new partner D, and how D acquires his share from A and B. We need to calculate the new profit-sharing ratio of all partners.

Understanding the Given Information

• Old partners: A, B, C
• Old profit-sharing ratio: 3:2:1
• New partner: D
• D's share of profit: \( \frac{1}{7} \)
• D gets his share: \( \frac{1}{3} \) from A and \( \frac{2}{3} \) from B. This means A gives up \( \frac{1}{3} \) of D's share, and B gives up \( \frac{2}{3} \) of D's share.

Calculating Partner Sacrifices

The portion of profit surrendered by old partners is their sacrifice. D's total share is \( \frac{1}{7} \).

• Amount sacrificed by A = \( \text{D's share} \times \text{Proportion from A} \)
• Amount sacrificed by A = \( \frac{1}{7} \times \frac{1}{3} = \frac{1 \times 1}{7 \times 3} = \frac{1}{21} \)

So, A sacrifices \( \frac{1}{21} \) of the total profit.

• Amount sacrificed by B = \( \text{D's share} \times \text{Proportion from B} \)
• Amount sacrificed by B = \( \frac{1}{7} \times \frac{2}{3} = \frac{1 \times 2}{7 \times 3} = \frac{2}{21} \)

So, B sacrifices \( \frac{2}{21} \) of the total profit.

C does not sacrifice any portion of profit to D as per the problem statement. C's sacrifice is 0.

Calculating New Profit Sharing Ratios

A partner's new share is calculated by subtracting their sacrifice from their old share.

The old shares are calculated from the old ratio 3:2:1. The total parts in the old ratio is \( 3 + 2 + 1 = 6 \).

• A's old share = \( \frac{3}{6} \)
• B's old share = \( \frac{2}{6} \)
• C's old share = \( \frac{1}{6} \)

Now, we calculate the new shares:

• A's new share = Old share - Sacrifice by A
• A's new share = \( \frac{3}{6} - \frac{1}{21} \)

To subtract these fractions, we find a common denominator for 6 and 21. The least common multiple is 42.

A's new share = \( \frac{3 \times 7}{6 \times 7} - \frac{1 \times 2}{21 \times 2} = \frac{21}{42} - \frac{2}{42} = \frac{21 - 2}{42} = \frac{19}{42} \)

• B's new share = Old share - Sacrifice by B
• B's new share = \( \frac{2}{6} - \frac{2}{21} \)

Using the common denominator 42:

B's new share = \( \frac{2 \times 7}{6 \times 7} - \frac{2 \times 2}{21 \times 2} = \frac{14}{42} - \frac{4}{42} = \frac{14 - 4}{42} = \frac{10}{42} \)

• C's new share = Old share - Sacrifice by C
• C's new share = \( \frac{1}{6} - 0 = \frac{1}{6} \)

To express C's share with the denominator 42:

C's new share = \( \frac{1 \times 7}{6 \times 7} = \frac{7}{42} \)

• D's share is given as \( \frac{1}{7} \).

To express D's share with the denominator 42:

D's share = \( \frac{1 \times 6}{7 \times 6} = \frac{6}{42} \)

Determining the New Profit Sharing Ratio

The new shares of A, B, C, and D are \( \frac{19}{42} \), \( \frac{10}{42} \), \( \frac{7}{42} \), and \( \frac{6}{42} \) respectively.

The new profit sharing ratio is the ratio of their new shares:

A : B : C : D = \( \frac{19}{42} : \frac{10}{42} : \frac{7}{42} : \frac{6}{42} \)

Multiplying by the common denominator (42) to remove fractions, we get the ratio:

New Ratio = 19 : 10 : 7 : 6

Let's verify that the new shares add up to 1:

\( \frac{19}{42} + \frac{10}{42} + \frac{7}{42} + \frac{6}{42} = \frac{19 + 10 + 7 + 6}{42} = \frac{42}{42} = 1 \)

The calculation is consistent.

Based on the calculation from the given information, the new profit sharing ratio is 19:10:7:6.

Comparing this with the given options:

• Option 1: 1:1:1:1
• Option 2: 9:5:4:6
• Option 3: 9:5:4:5
• Option 4: 9:5:6:6

The calculated ratio 19:10:7:6 is not listed among the options. However, the provided correct answer text is 9:5:4:6.

Let's state the provided correct answer.

Based on standard accounting principles and the specific information provided in the question, the calculated new profit-sharing ratio is 19:10:7:6.

Revision Table: Partnership Ratios

Additional Information on Partner Admission

The admission of a new partner is a reconstitution of the partnership firm. Besides the change in the profit-sharing ratio, other adjustments are often required, such as:

• Valuation and adjustment of Goodwill.
• Revaluation of assets and liabilities.
• Adjustment of accumulated profits and reserves.
• Adjustment of partners' capitals.

The method of calculating the new profit sharing ratio depends on how the new partner acquires their share. Common methods include:

• The new partner acquires a specific fraction of the total profit, and the old partners share the remaining profit in their old ratio.
• The new partner acquires their share by taking a specific fraction or amount from one or more existing partners' shares (as in this problem).
• The new partner acquires a share, and the old partners decide on a specific new ratio among themselves.
• The new partner brings in capital and goodwill, and the ratios are adjusted based on these contributions.

Understanding the specific wording in the problem ("gets 1/3 from A and 2/3 from B" vs. "A and B sacrifice in the ratio 1:2") is crucial for accurate calculation of the new profit sharing ratio and sacrifice ratio. In this problem, "gets 1/3 from A and 2/3 from B" usually implies \( \frac{1}{3} \) of D's share comes from A and \( \frac{2}{3} \) of D's share comes from B.