To solve the given optimization problem using Gurobi, we need to define the objective function and constraints based on the provided description.

The objective function is to minimize \(9.69 \times \text{hours worked by Mary} + 4.91 \times \text{hours worked by John} + 9.83 \times \text{hours worked by Hank}\).

Given constraints:
- Paperwork competence ratings for Mary, John, and Hank are 3, 7, and 6 respectively.
- Organization scores for Mary, John, and Hank are 1, 4, and 8 respectively.
- Minimum total paperwork competence rating from hours worked by Mary and Hank is 36.
- Minimum total paperwork competence rating from hours worked by all (Mary, John, Hank) is 36.
- Minimum total organization score from hours worked by John and Hank is 18.
- Minimum total organization score from hours worked by Mary and Hank is 16.
- Minimum total organization score from all (Mary, John, Hank) is 36.
- Additional constraints involving hours worked by each individual with specific coefficients.
- Maximum total paperwork competence rating from hours worked by Mary and John is 110.
- Maximum total paperwork competence rating from all (Mary, John, Hank) is 63.

Here's the Gurobi code to model this problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
hours_mary = m.addVar(lb=0, name="hours_worked_by_Mary")
hours_john = m.addVar(lb=0, name="hours_worked_by_John")
hours_hank = m.addVar(lb=0, name="hours_worked_by_Hank")

# Objective function
m.setObjective(9.69 * hours_mary + 4.91 * hours_john + 9.83 * hours_hank, GRB.MINIMIZE)

# Constraints
# Minimum total paperwork competence rating from Mary and Hank
m.addConstr(3 * hours_mary + 6 * hours_hank >= 36, name="paperwork_MH_min")

# Minimum total paperwork competence rating from all
m.addConstr(3 * hours_mary + 7 * hours_john + 6 * hours_hank >= 36, name="paperwork_all_min")

# Minimum total organization score from John and Hank
m.addConstr(4 * hours_john + 8 * hours_hank >= 18, name="organization_JH_min")

# Minimum total organization score from Mary and Hank
m.addConstr(hours_mary + 8 * hours_hank >= 16, name="organization_MH_min")

# Minimum total organization score from all
m.addConstr(hours_mary + 4 * hours_john + 8 * hours_hank >= 36, name="organization_all_min")

# Additional constraints
m.addConstr(-5 * hours_mary + 6 * hours_hank >= 0, name="additional_1")
m.addConstr(2 * hours_john - 8 * hours_hank >= 0, name="additional_2")

# Maximum total paperwork competence rating from Mary and John
m.addConstr(3 * hours_mary + 7 * hours_john <= 110, name="paperwork_MJ_max")

# Maximum total paperwork competence rating from all
m.addConstr(3 * hours_mary + 7 * hours_john + 6 * hours_hank <= 63, name="paperwork_all_max")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Mary: {hours_mary.x}")
    print(f"Hours worked by John: {hours_john.x}")
    print(f"Hours worked by Hank: {hours_hank.x}")
else:
    print("No optimal solution found.")

```