When a new partner is admitted into a partnership firm, the existing partners' profit sharing ratio changes. The new partner acquires a share of the future profits, which is usually taken from the old partners' shares. The question asks us to calculate the new profit sharing ratio of the partners after the admission of a new partner, Z.

Understanding the Old Profit Sharing Ratio

Initially, partners X and Y share profits and losses in a specific ratio. This is their old profit sharing ratio.

• Old partners: X and Y
• Old Profit Sharing Ratio: 3 ∶ 2

This means that out of a total of 5 parts (3 + 2), X gets 3 parts and Y gets 2 parts of the profit. So, X's share is $\frac{3}{5}$ and Y's share is $\frac{2}{5}$.

Admitting a New Partner

Z is admitted as a new partner. The share of profit given to the new partner is specified.

• New partner: Z
• Z's share in profits: $\frac{1}{5}$

Z will receive $\frac{1}{5}$ of the total profit of the firm.

Calculating the Remaining Share for Old Partners

After giving a share to the new partner Z, the remaining share of the profit is distributed among the old partners (X and Y) in their old profit sharing ratio. We assume that the old partners share the remaining profit in their old ratio unless specified otherwise.

Total profit of the firm is considered as 1 whole.

Share given to Z = $\frac{1}{5}$

Remaining share for X and Y = Total Share - Z's Share

Remaining share = $1 - \frac{1}{5}$

To subtract, we find a common denominator:

$1 = \frac{5}{5}$

Remaining share = $\frac{5}{5} - \frac{1}{5} = \frac{5 - 1}{5} = \frac{4}{5}$

So, the remaining share of profits to be shared by X and Y is $\frac{4}{5}$.

Calculating New Shares of Old Partners

X and Y will share this remaining $\frac{4}{5}$ share in their old ratio of 3 ∶ 2.

X's new share = X's old ratio in remaining share × Remaining share

X's old ratio (out of 5 parts) = $\frac{3}{5}$

X's new share = $\frac{3}{5} \times \frac{4}{5} = \frac{3 \times 4}{5 \times 5} = \frac{12}{25}$

Y's new share = Y's old ratio in remaining share × Remaining share

Y's old ratio (out of 5 parts) = $\frac{2}{5}$

Y's new share = $\frac{2}{5} \times \frac{4}{5} = \frac{2 \times 4}{5 \times 5} = \frac{8}{25}$

Determining the New Profit Sharing Ratio

The new profit sharing ratio is between all partners: X, Y, and Z. We have calculated their new shares:

• X's new share = $\frac{12}{25}$
• Y's new share = $\frac{8}{25}$
• Z's new share = $\frac{1}{5}$

To express the new profit sharing ratio, we need to have a common denominator for all shares. The denominators are 25, 25, and 5. The least common multiple is 25.

Convert Z's share to have a denominator of 25:

Z's share = $\frac{1}{5} = \frac{1 \times 5}{5 \times 5} = \frac{5}{25}$

Now, all shares have the same denominator:

• X's new share = $\frac{12}{25}$
• Y's new share = $\frac{8}{25}$
• Z's new share = $\frac{5}{25}$

The new profit sharing ratio is the ratio of the numerators when the denominators are the same.

New Ratio (X ∶ Y ∶ Z) = 12 ∶ 8 ∶ 5

Let's verify if the sum of the new shares equals 1:

Sum of shares = $\frac{12}{25} + \frac{8}{25} + \frac{5}{25} = \frac{12 + 8 + 5}{25} = \frac{25}{25} = 1$

This confirms our calculation is correct.

Summary of New Profit Sharing Ratio Calculation

Here is a step-by-step breakdown:

1. Identify old partners' ratio: X:Y = 3:2 ($\frac{3}{5}$, $\frac{2}{5}$).
2. Identify new partner's share: Z = $\frac{1}{5}$.
3. Calculate remaining share after Z's admission: $1 - \frac{1}{5} = \frac{4}{5}$.
4. Distribute remaining share among old partners in their old ratio:
   • X's new share = $\frac{3}{5} \times \frac{4}{5} = \frac{12}{25}$
   • Y's new share = $\frac{2}{5} \times \frac{4}{5} = \frac{8}{25}$
5. Combine new shares of all partners: X : Y : Z = $\frac{12}{25} : \frac{8}{25} : \frac{1}{5}$.
6. Convert shares to a common denominator (25): X : Y : Z = $\frac{12}{25} : \frac{8}{25} : \frac{5}{25}$.
7. Express as a ratio: 12 ∶ 8 ∶ 5.

The new profit sharing ratio of X, Y, and Z is 12 ∶ 8 ∶ 5.

Revision Table: Profit Sharing Ratios

This table summarises the ratios before and after Z's admission.

Additional Information: Sacrificing Ratio

When a new partner is admitted, the old partners usually sacrifice a portion of their share to accommodate the new partner's share. The ratio in which the old partners give up their shares is called the sacrificing ratio.

Sacrificing Ratio = Old Share - New Share

• X's Sacrifice = Old Share - New Share = $\frac{3}{5} - \frac{12}{25}$

To subtract, find a common denominator (25):

X's Old Share = $\frac{3}{5} = \frac{3 \times 5}{5 \times 5} = \frac{15}{25}$

X's Sacrifice = $\frac{15}{25} - \frac{12}{25} = \frac{3}{25}$

• Y's Sacrifice = Old Share - New Share = $\frac{2}{5} - \frac{8}{25}$

To subtract, find a common denominator (25):

Y's Old Share = $\frac{2}{5} = \frac{2 \times 5}{5 \times 5} = \frac{10}{25}$

Y's Sacrifice = $\frac{10}{25} - \frac{8}{25} = \frac{2}{25}$

The sacrificing ratio of X and Y is $\frac{3}{25} : \frac{2}{25}$, which is 3 ∶ 2. In this specific case, when the new partner's share is acquired from the old partners in their old profit sharing ratio, the sacrificing ratio is the same as the old profit sharing ratio.

The total sacrifice made by old partners ($\frac{3}{25} + \frac{2}{25} = \frac{5}{25}$) is equal to the new partner's share ($\frac{1}{5} = \frac{5}{25}$).