## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by Hank', and 'hours worked by John', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ : hours worked by Mary
- $x_2$ : hours worked by Hank
- $x_3$ : hours worked by John

## 3: Define the objective function in symbolic notation
The objective function to minimize is $5.15x_1 + 6.39x_2 + 4.5x_3$.

## 4: List the constraints in symbolic notation
The constraints are:
1. $14x_1 \leq 45$ (Mary's productivity rating constraint, but it's an equality in the sense that her rating is 14, so $14 = 14$ which doesn't limit $x_1$ directly but $x_1$ has an upper bound)
2. $8x_2 \leq 45$ (Hank's productivity rating constraint)
3. $16x_3 \leq 45$ (John's productivity rating constraint)
4. $14x_1 + 16x_3 \geq 8$
5. $8x_2 + 16x_3 \geq 10$
6. $14x_1 + 8x_2 + 16x_3 \geq 10$
7. $-9x_1 + 7x_3 \geq 0$
8. $5x_2 - 4x_3 \geq 0$
9. $8x_2 + 16x_3 \leq 25$
10. $14x_1 + 8x_2 + 16x_3 \leq 44$
11. $x_1$ is an integer (non-fractional)
12. $x_2$ is continuous (fractional allowed)
13. $x_3$ is an integer

## 5: Correcting and Clarifying Constraints
- The productivity ratings given are not constraints but parameters: $r_0 = {'x0': 14, 'x1': 8, 'x2': 16}$.
- The upper bound for each person's productivity rating times hours worked is 45, but this seems to be a misunderstanding since the ratings are fixed and not directly multiplied by hours in constraints.

## 6: Formulate the problem in Gurobi
Given the corrections and clarifications, let's formulate the problem:

## 7: Gurobi Code
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define variables
x1 = model.addVar(name="hours_worked_by_Mary", vtype=gurobi.GRB.INTEGER)  # Non-fractional hours worked by Mary
x2 = model.addVar(name="hours_worked_by_Hank", vtype=gurobi.GRB.CONTINUOUS)  # Fractional hours worked by Hank
x3 = model.addVar(name="hours_worked_by_John", vtype=gurobi.GRB.INTEGER)  # Integer hours worked by John

# Objective function
model.setObjective(5.15 * x1 + 6.39 * x2 + 4.5 * x3, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(14 * x1 <= 45, name="Mary_productivity")
model.addConstr(8 * x2 <= 45, name="Hank_productivity")
model.addConstr(16 * x3 <= 45, name="John_productivity")

model.addConstr(14 * x1 + 16 * x3 >= 8, name="Mary_John_min_productivity")
model.addConstr(8 * x2 + 16 * x3 >= 10, name="Hank_John_min_productivity")
model.addConstr(14 * x1 + 8 * x2 + 16 * x3 >= 10, name="Total_min_productivity")

model.addConstr(-9 * x1 + 7 * x3 >= 0, name="Mary_John_tradeoff")
model.addConstr(5 * x2 - 4 * x3 >= 0, name="Hank_John_tradeoff")

model.addConstr(8 * x2 + 16 * x3 <= 25, name="Hank_John_max_productivity")
model.addConstr(14 * x1 + 8 * x2 + 16 * x3 <= 44, name="Total_max_productivity")

# Optimize
model.optimize()

# Print solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objVal)
    print("Hours worked by Mary: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
    print("Hours worked by John: ", x3.varValue)
else:
    print("No optimal solution found")
```

## 8: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by Hank'), ('x3', 'hours worked by John')],
    'objective_function': '5.15*x1 + 6.39*x2 + 4.5*x3',
    'constraints': [
        '14*x1 <= 45',
        '8*x2 <= 45',
        '16*x3 <= 45',
        '14*x1 + 16*x3 >= 8',
        '8*x2 + 16*x3 >= 10',
        '14*x1 + 8*x2 + 16*x3 >= 10',
        '-9*x1 + 7*x3 >= 0',
        '5*x2 - 4*x3 >= 0',
        '8*x2 + 16*x3 <= 25',
        '14*x1 + 8*x2 + 16*x3 <= 44'
    ]
}
```