## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Ringo' and 'hours worked by Paul', which we can denote as $x_0$ and $x_1$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $9x_0 + 3x_1$.

## Step 3: List the constraints
The constraints given are:
- $4x_0 + 1x_1 \geq 22$
- $9x_0 + 1x_1 \geq 9$
- $5x_0 + 3x_1 \geq 28$
- $3x_0 + 4x_1 \geq 10$
- $-5x_0 + 10x_1 \geq 0$
- $4x_0 + 1x_1 \leq 36$
- $9x_0 + 1x_1 \leq 35$
- $5x_0 + 3x_1 \leq 38$
- $3x_0 + 4x_1 \leq 34$

## 4: Convert the problem into a Gurobi code
We will use Gurobi to solve this linear programming problem.

## 5: Write down the json representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Ringo'), ('x1', 'hours worked by Paul')],
    'objective_function': '9*x0 + 3*x1',
    'constraints': [
        '4*x0 + x1 >= 22',
        '9*x0 + x1 >= 9',
        '5*x0 + 3*x1 >= 28',
        '3*x0 + 4*x1 >= 10',
        '-5*x0 + 10*x1 >= 0',
        '4*x0 + x1 <= 36',
        '9*x0 + x1 <= 35',
        '5*x0 + 3*x1 <= 38',
        '3*x0 + 4*x1 <= 34'
    ]
}
```

## 6: Implement the problem in Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Ringo", lb=0)  # hours worked by Ringo
    x1 = model.addVar(name="hours_worked_by_Paul", lb=0)   # hours worked by Paul

    # Define the objective function
    model.setObjective(9 * x0 + 3 * x1, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(4 * x0 + x1 >= 22, name="computer_competence_rating")
    model.addConstr(9 * x0 + x1 >= 9, name="productivity_rating")
    model.addConstr(5 * x0 + 3 * x1 >= 28, name="paperwork_competence_rating")
    model.addConstr(3 * x0 + 4 * x1 >= 10, name="likelihood_to_quit_index")
    model.addConstr(-5 * x0 + 10 * x1 >= 0, name="ratio_constraint")
    model.addConstr(4 * x0 + x1 <= 36, name="computer_competence_rating_upper_bound")
    model.addConstr(9 * x0 + x1 <= 35, name="productivity_rating_upper_bound")
    model.addConstr(5 * x0 + 3 * x1 <= 38, name="paperwork_competence_rating_upper_bound")
    model.addConstr(3 * x0 + 4 * x1 <= 34, name="likelihood_to_quit_index_upper_bound")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Ringo: {x0.varValue}")
        print(f"Hours worked by Paul: {x1.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```