A and B are partners sharing profit in the ratio of 5:3. C is admitted to share future profits. Calculate the new profit sharing.

Steps to Solve

1. Understand the Initial Ratios

A and B share profits in the ratio of 5:3. To work with these ratios, we can express them in terms of a common total.

Let’s denote the profit shared by A as $5x$ and that of B as $3x$ where $x$ is a common multiplier.

2. Calculate Total Shares of Existing Partners

To find the total shares for partners A and B:

Total Shares = $5x + 3x = 8x$

3. Identify the New Partner's Share

Since C joins the partnership, we need to determine what the total shares will be with C included. The problem states the new profit sharing ratio will be $15:9:8$.

4. Calculate Share of Each Partner

The total parts in the new ratio $A:B:C = 15:9:8$ is:

Total Parts = $15 + 9 + 8 = 32$

Now,

• A's share = $\frac{15}{32}$
• B's share = $\frac{9}{32}$
• C's share = $\frac{8}{32}$

5. Calculate Sacrifice by Existing Partners

Next, we need to determine how much A and B sacrifice to accommodate C.

• A’s original share = $\frac{5}{8}$

• A’s new share = $\frac{15}{32}$

  To find A's sacrifice:

  $$ \text{Sacrifice by A} = \left(\frac{5}{8} - \frac{15}{32}\right) = \frac{20}{32} - \frac{15}{32} = \frac{5}{32} $$

• B’s original share = $\frac{3}{8}$

• B’s new share = $\frac{9}{32}$

  B's sacrifice is:

  $$ \text{Sacrifice by B} = \left(\frac{3}{8} - \frac{9}{32}\right) = \frac{12}{32} - \frac{9}{32} = \frac{3}{32} $$

6. Summary of Results

Finally, we establish the new profit sharing ratio:

The new ratio is $15:9:8$ for A, B, and C, respectively.