To tackle the given optimization problem, we first need to understand and translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the provided information.

### Symbolic Representation

- **Variables**:
  - Let $x_1$ represent 'hours worked by Bill'.
  - Let $x_2$ represent 'hours worked by Hank'.
  - Let $x_3$ represent 'hours worked by Peggy'.

- **Objective Function**: The goal is to maximize $1.88x_1 + 3.92x_2 + 4.99x_3$.

- **Constraints**:
  1. Work quality rating for Bill: $x_1 \cdot 3 = 3x_1$ (but since this is an equality based on a specific condition, we'll directly incorporate it into the model by ensuring $x_1$ is considered with its work quality rating in mind).
  2. Work quality rating for Hank: $x_2 \cdot 24 = 24x_2$.
  3. Work quality rating for Peggy: $x_3 \cdot 14 = 14x_3$.
  4. Combined work quality rating from Bill and Hank: $3x_1 + 24x_2 \leq 246$.
  5. Combined work quality rating from Hank and Peggy: $24x_2 + 14x_3 \leq 146$.
  6. Total combined work quality rating from all: $3x_1 + 24x_2 + 14x_3 \leq 146$.
  7. $x_1$ must be an integer (whole number of hours worked by Bill).
  8. $x_2$ must be an integer (integer amount of hours worked by Hank).
  9. $x_3$ can be any real number (non-integer amount of hours worked by Peggy).

### Symbolic Representation in JSON Format

```json
{
  'sym_variables': [('x1', 'hours worked by Bill'), ('x2', 'hours worked by Hank'), ('x3', 'hours worked by Peggy')],
  'objective_function': 'maximize 1.88*x1 + 3.92*x2 + 4.99*x3',
  'constraints': [
    '3*x1 + 24*x2 <= 246',
    '24*x2 + 14*x3 <= 146',
    '3*x1 + 24*x2 + 14*x3 <= 146',
    'x1 >= 0', 'x2 >= 0', 'x3 >= 0'  # Implicitly, hours worked cannot be negative
  ]
}
```

### Gurobi Code

To solve this optimization problem using Gurobi, we'll write the following Python code:

```python
from gurobipy import *

# Create a new model
m = Model("Work_Hours_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Bill")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Hank")
x3 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Peggy")

# Set the objective function
m.setObjective(1.88*x1 + 3.92*x2 + 4.99*x3, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x1 + 24*x2 <= 246, name="Bill_and_Hank_work_quality")
m.addConstr(24*x2 + 14*x3 <= 146, name="Hank_and_Peggy_work_quality")
m.addConstr(3*x1 + 24*x2 + 14*x3 <= 146, name="Total_work_quality")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Bill: {x1.x}")
    print(f"Hours worked by Hank: {x2.x}")
    print(f"Hours worked by Peggy: {x3.x}")
else:
    print("No optimal solution found.")
```