Explanation:

Solution and Explanation

Step 1: Understanding Gaining Ratio
The Gaining Ratio is calculated as:   

                                                            New Ratio−Old Ratio

Given Data: 

• Old Ratio: Rani: Sandhya: Kangana = 4:3:2
• New Ratio (after Rani retires): Sandhya: Kangana = 5:3

Step 2: Find Old Shares of Sandhya and Kangana
Total old share = 4+3+2=94 + 3 + 2 = 94+3+2=9

Sandhya's old share = 3/9

Kangana's old share = 2/9

Step 3: Find New Shares of Sandhya and Kangana
After Rani retires, the remaining two partners share profits in the ratio of 5:3.

Total new share = 5+3=85 + 3 = 85+3=8

Sandhya's new share = 5/8

Kangana's new share = 3/8

Step 4: Calculate Gaining Ratio

                                                 Gaining Share=New Share−Old Share

For Sandhya:        5    -   3    =   45 - 24    =   21    

                             8         9          72                72

For Kangana:         3    -   2    =    27 - 16             =  11    

                               8       9               72                     72

Thus, the Gaining Ratio = 21:11

Correct Answer: (B) 21:11