## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary' and 'hours worked by Paul', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $4.13x_1 + 9.68x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $6.58x_1 + 2.37x_2 \geq 14$
- $6.32x_1 + 2.73x_2 \geq 31$
- $-9x_1 + x_2 \geq 0$
- $6.58x_1 + 2.37x_2 \leq 28$
- $6.32x_1 + 2.73x_2 \leq 46$
- $x_1$ is an integer
- $x_2$ is an integer

## 4: Convert the problem into a Gurobi code
We will use Gurobi's Python API to model and solve this problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="hours_worked_by_Mary", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Paul", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(4.13 * x1 + 9.68 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(6.58 * x1 + 2.37 * x2 >= 14, name="likelihood_to_quit_index_constraint")
m.addConstr(6.32 * x1 + 2.73 * x2 >= 31, name="computer_competence_rating_constraint")
m.addConstr(-9 * x1 + x2 >= 0, name="hours_worked_constraint")
m.addConstr(6.58 * x1 + 2.37 * x2 <= 28, name="likelihood_to_quit_index_upper_bound_constraint")
m.addConstr(6.32 * x1 + 2.73 * x2 <= 46, name="computer_competence_rating_upper_bound_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Mary: {x1.varValue}")
    print(f"Hours worked by Paul: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by Paul')],
    'objective_function': '4.13*x1 + 9.68*x2',
    'constraints': [
        '6.58*x1 + 2.37*x2 >= 14',
        '6.32*x1 + 2.73*x2 >= 31',
        '-9*x1 + x2 >= 0',
        '6.58*x1 + 2.37*x2 <= 28',
        '6.32*x1 + 2.73*x2 <= 46',
        'x1 is an integer',
        'x2 is an integer'
    ]
}
```