## Problem Description and Symbolic Representation

The problem involves a summer painting company that needs to minimize its wage bill by determining the optimal number of students and full-time employees to hire. The company has the following requirements:

- At least 100 painters are needed.
- At least 30 painters must be full-time employees.
- The number of full-time employees should be at least half the number of students.

Let's denote:
- $x_1$ as the number of students.
- $x_2$ as the number of full-time employees.

The objective is to minimize the wage bill, which can be represented as $200x_1 + 500x_2$.

The constraints based on the problem description are:
1. $x_1 + x_2 \geq 100$ (at least 100 painters).
2. $x_2 \geq 30$ (at least 30 full-time employees).
3. $x_2 \geq 0.5x_1$ (number of full-time employees should be at least half the number of students).
4. $x_1 \geq 0$ and $x_2 \geq 0$ (non-negativity constraints, as the number of employees cannot be negative).

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'number of students'), ('x2', 'number of full-time employees')],
    'objective_function': '200*x1 + 500*x2',
    'constraints': [
        'x1 + x2 >= 100',
        'x2 >= 30',
        'x2 >= 0.5*x1',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, name="students")  # number of students
    x2 = model.addVar(lb=0, name="full_time_employees")  # number of full-time employees

    # Objective function: minimize the wage bill
    model.setObjective(200 * x1 + 500 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 + x2 >= 100, name="total_painters")
    model.addConstr(x2 >= 30, name="min_full_time_employees")
    model.addConstr(x2 >= 0.5 * x1, name="full_time_to_students_ratio")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Number of students: {x1.varValue}")
        print(f"Number of full-time employees: {x2.varValue}")
        print(f"Minimum wage bill: ${model.objVal:.2f}")
    else:
        print("The model is infeasible.")

solve_painting_company_problem()
```