## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Jean' and 'hours worked by Ringo'. Let's denote 'hours worked by Jean' as $x_1$ and 'hours worked by Ringo' as $x_2$. The objective function to minimize is $3.51x_1 + 7.43x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
- $6x_1 + 21x_2 \geq 44$
- $13x_1 + 10x_2 \geq 46$
- $9x_1 + 8x_2 \geq 16$
- $3x_1 - 4x_2 \geq 0$
- $6x_1 + 21x_2 \leq 113$
- $13x_1 + 10x_2 \leq 126$
- $9x_1 + 8x_2 \leq 65$

## 3: Provide the symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Ringo')],
'objective_function': '3.51*x1 + 7.43*x2',
'constraints': [
    '6*x1 + 21*x2 >= 44',
    '13*x1 + 10*x2 >= 46',
    '9*x1 + 8*x2 >= 16',
    '3*x1 - 4*x2 >= 0',
    '6*x1 + 21*x2 <= 113',
    '13*x1 + 10*x2 <= 126',
    '9*x1 + 8*x2 <= 65'
]
}
```

## 4: Convert the problem into Gurobi code
Now, let's write the Gurobi code for this problem:
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Jean', lb=0)  # No lower bound specified, assuming 0
    x2 = model.addVar(name='hours_worked_by_Ringo', lb=0)  # No lower bound specified, assuming 0

    # Define the objective function
    model.setObjective(3.51 * x1 + 7.43 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(6 * x1 + 21 * x2 >= 44)
    model.addConstr(13 * x1 + 10 * x2 >= 46)
    model.addConstr(9 * x1 + 8 * x2 >= 16)
    model.addConstr(3 * x1 - 4 * x2 >= 0)
    model.addConstr(6 * x1 + 21 * x2 <= 113)
    model.addConstr(13 * x1 + 10 * x2 <= 126)
    model.addConstr(9 * x1 + 8 * x2 <= 65)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Jean: {x1.varValue}')
        print(f'Hours worked by Ringo: {x2.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```