R, S, and K are partners sharing profits in the ratio of $4:3:2$. R retires, and S and K decide to share future profits in the ratio of $5:3$. Calculate their gaining ratio.

Let us compute the gaining ratio using the formula:
$$\text{Gaining Share}=\text{New Share}-\text{Old Share}$$
Identify the Old Shares:
$$\text{Old Share of S}=\frac{3}{9},\text{Old Share of K}=\frac{2}{9}$$
Identify the New Shares:
$$\text{New Share of S}=\frac{5}{8},\text{New Share of K}=\frac{3}{8}$$
Calculate the Gain for Each Partner:For S:
$$\text{Gain of S}=\frac{5}{8}-\frac{3}{9}=\frac{45-24}{72}=\frac{21}{72}$$
For K:
$$\text{Gain of K}=\frac{3}{8}-\frac{2}{9}=\frac{27-16}{72}=\frac{11}{72}$$
Determine the Gaining Ratio:Comparing their gains ($\frac{21}{72}:\frac{11}{72}$), we get a ratio of $21:11$.