A, B and C are partners in a firm sharing profits and losses in the ratio of 4 : 3 : 2. They agreed to take D into partnership and gave him $${\frac{1}{8}^{{\text{th}}}}$$ share. What will be their new profit sharing ratio?

The correct answer is $\boxed{\text{B. 28 : 21 : 14 : 9}}$.

The original profit sharing ratio of A, B, and C is 4 : 3 : 2. This means that A gets 4 parts of the profit, B gets 3 parts, and C gets 2 parts. When D is added to the partnership, he gets $\frac{1}{8}$ of the profit. This means that the new profit sharing ratio is 4 + $\frac{1}{8}$ : 3 + $\frac{1}{8}$ : 2 + $\frac{1}{8}$ = 28 : 21 : 14 : 9.

Here is a step-by-step solution:

1. Find the total number of parts in the new profit sharing ratio. This is done by adding the original number of parts for each partner. In this case, the total number of parts is $4 + 3 + 2 = 9$.
2. Find the new profit sharing ratio for each partner by multiplying their original profit sharing ratio by the total number of parts. In this case, the new profit sharing ratio for A is $4 \times 9 = 36$, the new profit sharing ratio for B is $3 \times 9 = 27$, the new profit sharing ratio for C is $2 \times 9 = 18$, and the new profit sharing ratio for D is $\frac{1}{8} \times 9 = 1$.
3. Express the new profit sharing ratio in a simplified form. In this case, the new profit sharing ratio is 28 : 21 : 14 : 9.