The correct answer is $\boxed{\text{D}}$.
Initially, R and S share the profits in the ratio of 5 : 3. This means that R gets 5 parts of the profit and S gets 3 parts of the profit.
When T joins the firm, R gives $\frac{1}{4}$ of his share to T and S gives $\frac{2}{5}$ of his share to T. This means that R now gets $\frac{3}{4}$ of his original share and S now gets $\frac{3}{5}$ of his original share.
The total amount of profit that R, S, and T now share is 5 + 3 + 1 = 9 parts.
R’s share of the profit is now $\frac{3}{4} \times 5 = \frac{15}{4}$ parts.
S’s share of the profit is now $\frac{3}{5} \times 3 = \frac{9}{5}$ parts.
T’s share of the profit is $\frac{9}{9} = 1$ part.
Therefore, the new profit sharing ratio of R, S, and T is $\frac{15}{4} : \frac{9}{5} : 1 = 20 : 9 : 11$.
Here is a brief explanation of each option:
- Option A: This option is incorrect because it does not take into account the fact that R and S give some of their shares to T.
- Option B: This option is incorrect because it does not take into account the fact that R and S give different amounts of their shares to T.
- Option C: This option is incorrect because it does not take into account the fact that the total amount of profit that R, S, and T now share is 9 parts.
- Option D: This option is correct because it takes into account all of the factors mentioned above.