Understanding the Partnership Change

In this problem, Adesh and Bhadresh are partners who decide to admit a new partner, Chandresh. When a new partner is admitted, the existing profit-sharing ratio among the old partners changes because the new partner will also share in the profits.

We are given the old profit-sharing ratio between Adesh and Bhadresh and details about how Chandresh acquires his share. We need to calculate the new profit-sharing ratio among Adesh, Bhadresh, and Chandresh.

Initial Profit Sharing Ratio

Adesh and Bhadresh share profits and losses in the ratio of 3 ∶ 2.

• Adesh's old share = $\frac{3}{3+2} = \frac{3}{5}$
• Bhadresh's old share = $\frac{2}{3+2} = \frac{2}{5}$

Chandresh's Admission and Share Acquisition

Chandresh is admitted for a 20% share of the profits.

• Chandresh's share = $20\%$ of total profits = $\frac{20}{100} = \frac{1}{5}$

This $\frac{1}{5}$ share is acquired in a specific way:

• Half of Chandresh's share is gifted by Adesh.
• The remaining half is acquired by Chandresh from Adesh and Bhadresh in equal proportion.

Calculating Share Gifted by Adesh

Half of Chandresh's share is gifted by Adesh.

• Share gifted by Adesh = $\frac{1}{2} \times$ Chandresh's share = $\frac{1}{2} \times \frac{1}{5} = \frac{1}{10}$

This $\frac{1}{10}$ is Adesh's direct sacrifice.

Calculating Remaining Share Acquired

The remaining share is acquired from Adesh and Bhadresh equally.

• Remaining share for Chandresh = Chandresh's total share - Share gifted by Adesh = $\frac{1}{5} - \frac{1}{10}$
• To subtract, find a common denominator (10): $\frac{2}{10} - \frac{1}{10} = \frac{1}{10}$

This remaining $\frac{1}{10}$ share is acquired equally from Adesh and Bhadresh.

• Share acquired from Adesh = $\frac{1}{2} \times$ Remaining share = $\frac{1}{2} \times \frac{1}{10} = \frac{1}{20}$
• Share acquired from Bhadresh = $\frac{1}{2} \times$ Remaining share = $\frac{1}{2} \times \frac{1}{10} = \frac{1}{20}$

Calculating Sacrifices by Old Partners

The total sacrifice made by each old partner is the amount of their share given up for the new partner.

• Total sacrifice by Adesh = Share gifted by Adesh + Share acquired from Adesh = $\frac{1}{10} + \frac{1}{20}$
• To add, find a common denominator (20): $\frac{2}{20} + \frac{1}{20} = \frac{3}{20}$
• Total sacrifice by Bhadresh = Share acquired from Bhadresh = $\frac{1}{20}$

Calculating New Profit Sharing Ratio

The new share of each old partner will be their old share minus their sacrifice.

• Adesh's new share = Adesh's old share - Adesh's sacrifice = $\frac{3}{5} - \frac{3}{20}$
• To subtract, find a common denominator (20): $\frac{12}{20} - \frac{3}{20} = \frac{9}{20}$

• Bhadresh's new share = Bhadresh's old share - Bhadresh's sacrifice = $\frac{2}{5} - \frac{1}{20}$
• To subtract, find a common denominator (20): $\frac{8}{20} - \frac{1}{20} = \frac{7}{20}$

Chandresh's share is already known:

• Chandresh's share = $\frac{1}{5}$

To express the new ratio, we need a common denominator for all shares (20).

• Adesh's new share = $\frac{9}{20}$
• Bhadresh's new share = $\frac{7}{20}$
• Chandresh's new share = $\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$

The new profit-sharing ratio is the ratio of these new shares:

Adesh ∶ Bhadresh ∶ Chandresh = $\frac{9}{20} : \frac{7}{20} : \frac{4}{20}$

Multiplying by the common denominator (20) gives the ratio:

9 ∶ 7 ∶ 4

Comparing with Given Options

Let's compare our calculated new profit-sharing ratio (9 ∶ 7 ∶ 4) with the given options:

The calculated new profit-sharing ratio of 9 ∶ 7 ∶ 4 does not match any of the first three options. Therefore, the correct choice is that none of the provided options is correct.