Step 1: Initial Ratio of Contributions
The initial ratio of contributions among Rana, Sana, and Kamana was 4:3:2, totaling 9 parts. \[\text{Sana's initial share} = \frac{3}{9} = \frac{1}{3}, \text{Kamana's initial share} = \frac{2}{9} \]
Step 2: Revised Ratio After Rana's Retirement
Following Rana's retirement, the new ratio of contributions between Sana and Kamana became 5:3, totaling 8 parts. \[\text{Sana's revised share} = \frac{5}{8}, \text{Kamana's revised share} = \frac{3}{8} \]
Step 3: Calculation of Gains
\[\text{Sana's gain} = \frac{5}{8} - \frac{1}{3} = \frac{15 - 8}{24} = \frac{7}{24} \] \[\text{Kamana's gain} = \frac{3}{8} - \frac{2}{9} = \frac{27 - 16}{72} = \frac{11}{72} \]
Step 4: Expressing Gains as a Ratio
To express the gains in a common ratio, the Least Common Multiple (LCM) of the denominators is calculated, which is 72. \[\text{Sana's gain} = \frac{7}{24} = \frac{21}{72}, \text{Kamana's gain} = \frac{11}{72} \] The resulting ratio of gains is 21 : 11.
Final Answer: \[\boxed{\text{Gaining Ratio = 21:11}} \]