Question

X, Y and Z share profits in the ratio of 4:3:2. Y retires and his share is acquired by X and Z in the ratio of 2:1.
The gaining ratio of X and Z is:

• A. 2 : 1
• B. 1 : 2
• C. 4 : 3
• D. 3 : 1

Hint:

In retirement problems, the ratio in which continuing partners acquire the retiring partner's share is always identical to their gaining ratio.
You do not need to carry out the calculation of individual gains to find the gaining ratio.
The ratio given for the acquisition of the retiring partner's share (2:1 in this case) is directly the gaining ratio.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem focuses on the retirement of a partner and the calculation of the gaining ratio of the remaining partners.
The existing partners are X, Y, and Z, and they share profits in the ratio of 4:3:2.
Partner Y retires from the firm.
The retiring partner's share is acquired by the remaining partners, X and Z, in a specific proportion of 2:1.
We need to determine the gaining ratio of X and Z.

Step 2: Key Formula or Approach:
1. Gaining ratio is the ratio in which the continuing partners acquire the share of profit from the retiring partner.
2. If the problem states the specific ratio in which the continuing partners acquire the retiring partner's share, that acquisition ratio is itself the gaining ratio.
3. Let us confirm this mathematically by calculating the individual gains:
\[ \text{Individual Gain} = \text{Retiring Partner's Share} \times \text{Acquisition Share} \]
\[ \text{Gaining Ratio} = \text{Gain of X} : \text{Gain of Z} \]

Step 3: Detailed Explanation:
1. We begin with the old profit-sharing ratio of X, Y, and Z, which is 4:3:2.
The individual shares of the partners are:
\[ \text{X's Old Share} = \frac{4}{9} \]
\[ \text{Y's Old Share} = \frac{3}{9} \]
\[ \text{Z's Old Share} = \frac{2}{9} \]
2. Partner Y retires, which means his share of \(\frac{3}{9}\) is distributed between X and Z.
3. The problem states that X and Z acquire Y's share in the ratio of 2:1.
4. Let us calculate the actual gain for X:
\[ \text{Gain of X} = \frac{3}{9} \times \frac{2}{3} = \frac{6}{27} \]
5. Now, let us calculate the actual gain for Z:
\[ \text{Gain of Z} = \frac{3}{9} \times \frac{1}{3} = \frac{3}{27} \]
6. Comparing the gains of X and Z to determine the gaining ratio:
\[ \text{Gaining Ratio} = \text{Gain of X} : \text{Gain of Z} = \frac{6}{27} : \frac{3}{27} \]
This simplifies to:
\[ \text{Gaining Ratio} = 6 : 3 = 2 : 1 \]
This mathematical proof confirms that the gaining ratio is exactly the same as the ratio in which they acquired the retiring partner's share.

Step 4: Final Answer:
The gaining ratio of X and Z is 2:1.
Therefore, Option (A) is the correct answer.