Let's denote the total number of candies as $T$.
There are 4 people: Alice (A), Ben (B), Cane (C), and Dino (D).

Step 1: Initial Distribution
The candies are distributed equally among the four people.
- Alice: $\(A_0=\frac{T}{4}\)$

- Ben: $\(B_0=\frac{T}{4}\)$

- Cane: $\(C_0=\frac{T}{4}\)$

- Dino: $\(D_0=\frac{T}{4}\)$

Step 2: Alice Distributes Her Candies

Alice distributes $\(\frac{3}{4}\)$ of her candies equally among the other three people (Ben, Cane, and Dino).

The amount Alice distributes is $\(\frac{3}{4} \times A_0=\frac{3}{4} \times \frac{T}{4}=\frac{3 T}{16}\)$.

This amount is divided equally among 3 people, so each receives: $\(\frac{3 T}{16} \div 3=\frac{T}{16}\)$.


Let's update their candy counts:

- Alice's remaining candies: $\(A_1=A_0-\frac{3}{4} A_0=\frac{1}{4} A_0=\frac{1}{4} \times \frac{T}{4}=\frac{T}{16}\)$

- Ben's candies: $\(B_1=B_0+\frac{T}{16}=\frac{T}{4}+\frac{T}{16}=\frac{4 T}{16}+\frac{T}{16}=\frac{5 T}{16}\)$

- Cane's candies: $\(C_1=C_0+\frac{T}{16}=\frac{T}{4}+\frac{T}{16}=\frac{4 T}{16}+\frac{T}{16}=\frac{5 T}{16}\)$

- Dino's candies: $\(D_1=D_0+\frac{T}{16}=\frac{T}{4}+\frac{T}{16}=\frac{4 T}{16}+\frac{T}{16}=\frac{5 T}{16}\)$

Step 3: Ben Gives Candies to Dino

Ben gives half of his current candies to Dino.


The amount Ben gives to Dino is $\(\frac{1}{2} \times B_1=\frac{1}{2} \times \frac{5 T}{16}=\frac{5 T}{32}\)$.

Let's update their final candy counts:

- Alice's final candies: $\(A_2=\frac{T}{16}\)$

- Ben's final candies: $\(B_2=B_1-\frac{5 T}{32}=\frac{5 T}{16}-\frac{5 T}{32}=\frac{10 T}{32}-\frac{5 T}{32}=\frac{5 T}{32}\)$

- Cane's final candies: $\(C_2=\frac{5 T}{16}\)$

- Dino's final candies: $\(D_2=D_1+\frac{5 T}{32}=\frac{5 T}{16}+\frac{5 T}{32}=\frac{10 T}{32}+\frac{5 T}{32}=\frac{15 T}{32}\)$

Final Question:
What fraction of the total number of candies does Dino have in the end?

Dino has $\(\frac{15 T}{32}\)$ candies. The total number of candies is $T$.

The fraction is $\(\frac{15 T / 32}{T}=\frac{15}{32}\)$.

The final answer is $\(\frac{15}{32}\)$.