The correct answer is $\boxed{\frac{14}{42}}$.
Initially, X and Y share profits in the ratio of 4:3. This means that X receives 4 parts of the profit and Y receives 3 parts of the profit. When Z is admitted into the partnership, he receives a $\frac{1}{3}$ share of the profits. This means that the total number of shares in the partnership is now 12, with X and Y each receiving 4 shares and Z receiving 4 shares. If Z receives his share from X and Y in equal proportion, then X and Y each receive 2 shares from Z. This means that X now has a total of 6 shares in the partnership, Y now has a total of 5 shares in the partnership, and Z has a total of 4 shares in the partnership. Therefore, Y’s share of the profits in the future will be $\frac{5}{12}$, or $\boxed{\frac{14}{42}}$.
Here is a breakdown of each option:
- Option A: $\frac{11}{42}$. This is the share of X in the profits in the future. This is incorrect because X’s share is $\frac{4}{12}$, not $\frac{11}{42}$.
- Option B: $\frac{17}{42}$. This is the share of Z in the profits in the future. This is incorrect because Z’s share is $\frac{4}{12}$, not $\frac{17}{42}$.
- Option C: $\frac{14}{42}$. This is the correct answer. This is because Y’s share is $\frac{5}{12}$, which is equal to $\frac{14}{42}$.
- Option D: $\frac{28}{42}$. This is the share of X and Y in the profits in the future combined. This is incorrect because X and Y’s combined share is $\frac{8}{12}$, not $\frac{28}{42}$.