## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation.

Let's define the symbolic variables:
- `x1`: consoles
- `x2`: discs

The objective is to maximize profit. The profit from selling `x1` consoles at a profit of $200 each and `x2` discs at a profit of $30 each can be represented as:
- Objective function: `200*x1 + 30*x2`

The constraints given are:
- The total cost is at most $30000: `300*x1 + 30*x2 <= 30000`
- A minimum of 20 but at most 50 consoles are sold: `20 <= x1 <= 50`
- The number of discs sold is at most five times the number of consoles sold: `x2 <= 5*x1`
- Non-negativity constraints: `x1 >= 0, x2 >= 0`

However, since `x1` and `x2` represent quantities of items, we can reasonably assume they must be integers.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'consoles'), ('x2', 'discs')],
    'objective_function': '200*x1 + 30*x2',
    'constraints': [
        '300*x1 + 30*x2 <= 30000',
        '20 <= x1 <= 50',
        'x2 <= 5*x1',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=0, ub=50, vtype=gurobi.GRB.INTEGER, name="consoles")
    x2 = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="discs")

    # Objective function: maximize profit
    model.setObjective(200*x1 + 30*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(300*x1 + 30*x2 <= 30000, name="budget_constraint")
    model.addConstr(x1 >= 20, name="min_consoles_constraint")
    model.addConstr(x2 <= 5*x1, name="discs_to_consoles_ratio_constraint")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Consoles: {x1.varValue}")
        print(f"Discs: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible or unbounded.")

solve_optimization_problem()
```