Calculating New Profit Sharing Ratio in Partnership

When a new partner is admitted into a business, the existing partners' profit sharing ratio changes. The new partner gets a share of the future profits, which reduces the share available for the old partners. The new profit sharing ratio determines how all partners, including the new one, will share profits or losses going forward.

Understanding the Problem

We are given:

• Old partners: X and Y.
• Old profit sharing ratio of X and Y: 3 : 2. This means X gets \(\frac{3}{3+2} = \frac{3}{5}\) of the profit shared between X and Y, and Y gets \(\frac{2}{3+2} = \frac{2}{5}\).
• New partner: Z.
• Z's share in the total profit: \(\frac{1}{5}\).

We need to calculate the new profit sharing ratio among X, Y, and Z.

Step-by-Step Calculation of New Profit Sharing Ratio

The calculation of the new profit sharing ratio depends on how the new partner acquires their share. In this case, it's a simple admission where the new partner takes a specific share from the total profit, and the old partners share the remaining profit in their old ratio.

Step 1: Calculate the Remaining Share of Profit

Assume the total profit of the firm is represented by 1.

Z's share of profit is \(\frac{1}{5}\).

The remaining share of profit that will be shared among the old partners (X and Y) is calculated as:

Remaining Share = Total Profit - Z's Share

Remaining Share = \(1 - \frac{1}{5}\)

To subtract, we find a common denominator:

Remaining Share = \(\frac{5}{5} - \frac{1}{5} = \frac{5-1}{5} = \frac{4}{5}\)

So, \(\frac{4}{5}\) of the total profit is available for X and Y to share.

Step 2: Calculate X's New Share of Profit

X will get his old share (\(\frac{3}{5}\)) of the remaining profit (\(\frac{4}{5}\)).

X's New Share = X's Old Share \(\times\) Remaining Share

X's New Share = \(\frac{3}{5} \times \frac{4}{5}\)

X's New Share = \(\frac{3 \times 4}{5 \times 5} = \frac{12}{25}\)

Step 3: Calculate Y's New Share of Profit

Y will get his old share (\(\frac{2}{5}\)) of the remaining profit (\(\frac{4}{5}\)).

Y's New Share = Y's Old Share \(\times\) Remaining Share

Y's New Share = \(\frac{2}{5} \times \frac{4}{5}\)

Y's New Share = \(\frac{2 \times 4}{5 \times 5} = \frac{8}{25}\)

Step 4: State Z's Share and Find a Common Denominator

Z's share is given as \(\frac{1}{5}\).

To express Z's share with the same denominator as X's and Y's new shares (which is 25), we multiply the numerator and the denominator by 5:

Z's Share = \(\frac{1}{5} = \frac{1 \times 5}{5 \times 5} = \frac{5}{25}\)

Step 5: Determine the New Profit Sharing Ratio

The new profit sharing ratio is the ratio of the new shares of X, Y, and Z.

New Ratio (X : Y : Z) = X's New Share : Y's New Share : Z's Share

New Ratio = \(\frac{12}{25} : \frac{8}{25} : \frac{5}{25}\)

Since the denominators are the same, the new profit sharing ratio is the ratio of the numerators.

New Profit Sharing Ratio = 12 : 8 : 5

Summary of New Profit Sharing Ratio

The new profit sharing ratio among X, Y, and Z is 12 : 8 : 5.

The ratio 12 : 8 : 5 represents the proportions in which X, Y, and Z will share future profits or losses of the partnership.

Revision Table: Partnership Profit Ratios

Additional Information: Sacrifice Ratio

When a new partner is admitted, the old partners usually have to give up (sacrifice) a portion of their share of profits to accommodate the new partner. The ratio in which the old partners sacrifice their share is called the Sacrifice Ratio.

Sacrifice Ratio = Old Share - New Share

• X's Sacrifice = Old Share of X - New Share of X = \(\frac{3}{5} - \frac{12}{25}\)
• To subtract, find a common denominator (25): \(\frac{3 \times 5}{5 \times 5} - \frac{12}{25} = \frac{15}{25} - \frac{12}{25} = \frac{3}{25}\)
• Y's Sacrifice = Old Share of Y - New Share of Y = \(\frac{2}{5} - \frac{8}{25}\)
• To subtract, find a common denominator (25): \(\frac{2 \times 5}{5 \times 5} - \frac{8}{25} = \frac{10}{25} - \frac{8}{25} = \frac{2}{25}\)

The Sacrifice Ratio of X and Y is \(\frac{3}{25} : \frac{2}{25}\), which simplifies to 3 : 2.

In this specific case, where the old partners share the remaining profit in their old ratio, the Sacrifice Ratio is the same as the Old Profit Sharing Ratio (3:2). This is a common scenario in partnership accounting when calculating the new profit sharing ratio upon admission of a partner.