Question

Geetha, Latha and Sangeetha are partners sharing profits and losses in the ratio of \(5 : 3 : 2\) respectively. Latha retires from the firm. Geetha and Sangeetha decided to share future profits in the ratio of \(3 : 2\).
Calculate Gaining Ratio of Geetha and Sangeetha.

Hint:

To verify your calculation, add the individual gains: \(\frac{1}{10} + \frac{2}{10} = \frac{3}{10}\). This should exactly equal the retiring partner's (Latha's) old share, which was \(\frac{3}{10}\).

Answer 1

Step 1: Understanding the Concept:
When a partner retires, the remaining partners acquire the retiring partner's share. This increases their original profit-sharing ratio. The proportion in which the remaining partners acquire this share is called the Gaining Ratio.

Step 2: Key Formula or Approach:
\[ \text{Gaining Ratio} = \text{New Ratio} - \text{Old Ratio} \]

Step 2: Detailed Explanation:
Given Data:
Old Ratio of Geetha, Latha, and Sangeetha = \(5 : 3 : 2\).
Therefore, Old Share of Geetha = \(\frac{5}{10}\)
Old Share of Sangeetha = \(\frac{2}{10}\)
Latha retires.
New Ratio of Geetha and Sangeetha = \(3 : 2\).
Therefore, New Share of Geetha = \(\frac{3}{5}\)
New Share of Sangeetha = \(\frac{2}{5}\)
Now, applying the formula:
Gain of Geetha = New Share \(-\) Old Share
\( = \frac{3}{5} - \frac{5}{10} \)
\( = \frac{6 - 5}{10} = \frac{1}{10} \)
Gain of Sangeetha = New Share \(-\) Old Share
\( = \frac{2}{5} - \frac{2}{10} \)
\( = \frac{4 - 2}{10} = \frac{2}{10} \)
The gaining ratio is the ratio of their individual gains.
Gaining Ratio of Geetha and Sangeetha = \(\frac{1}{10} : \frac{2}{10} = 1 : 2\).

Step 3: Final Answer:
The Gaining Ratio of Geetha and Sangeetha is \(1 : 2\).