Rana, Sana and Kamana are partners, sharing profits in the ratio 4:3:2. Rana retires; Sana and Kamana decided to share profits in the future in the ratio of 5:3. The Gaining Ratio of Sana and Kamana will be

The correct option is 21:11.

Step-by-Step Explanation:

To find the gaining ratio of the remaining partners (Sana and Kamana) after Rana's retirement, we use the following formula:
$$\text{Gaining Share}=\text{New Share}-\text{Old Share}$$

1. Identify the Old Ratio and Old Shares:
The old profit-sharing ratio among Rana, Sana, and Kamana is 4:3:2.
Therefore, their old shares are:
Rana's old share = $\frac{4}{4+3+2}=\frac{4}{9}$
Sana's old share = $\frac{3}{9}$
Kamana's old share = $\frac{2}{9}$

2. Identify the New Ratio and New Shares:
After Rana retires, the new profit-sharing ratio between Sana and Kamana is decided as 5:3.
Therefore, their new shares are:
Sana's new share = $\frac{5}{5+3}=\frac{5}{8}$
Kamana's new share = $\frac{3}{8}$

3. Calculate the Gaining Share for each remaining partner:

For Sana:
$$\text{Sana's Gain}=\frac{5}{8}-\frac{3}{9}$$
Taking the common denominator of 8 and 9, which is 72:
$$\text{Sana's Gain}=\frac{5\times 9-3\times 8}{72}=\frac{45-24}{72}=\frac{21}{72}$$

For Kamana:
$$\text{Kamana's Gain}=\frac{3}{8}-\frac{2}{9}$$
Taking the common denominator of 72:
$$\text{Kamana's Gain}=\frac{3\times 9-2\times 8}{72}=\frac{27-16}{72}=\frac{11}{72}$$

4. Determine the Gaining Ratio:
Comparing the gains of Sana and Kamana:
$$\text{Gaining Ratio}=\frac{21}{72}:\frac{11}{72}=21:11$$

Hence, the gaining ratio of Sana and Kamana is 21:11.