## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Jean', and 'hours worked by Mary'. Let's denote them 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 Hank
- $x_2$ : hours worked by Jean
- $x_3$ : hours worked by Mary

## 3: Define the objective function in symbolic notation
The objective function to minimize is $6.35x_1 + 4.82x_2 + 7.75x_3$.

## 4: List the constraints in symbolic notation
The constraints are:
- $5.9x_1 = 5.9$ (Hank's organization score is 5.9, but this seems to be an equality constraint directly given, not requiring multiplication by $x_1$)
- $19.39x_2 = 19.39$ (Jean's organization score is 19.39, similar to above)
- $20.41x_3 = 20.41$ (Mary's organization score is 20.41, similar to above)
- $5.9x_1 + 20.41x_3 \geq 87$
- $5.9x_1 + 19.39x_2 \geq 61$
- $5.9x_1 + 19.39x_2 + 20.41x_3 \geq 61$
- $-10x_1 + 6x_3 \geq 0$
- $7x_2 - 9x_3 \geq 0$
- $5.9x_1 + 19.39x_2 \leq 175$

However, upon reevaluation, it seems the first three constraints might actually be directly assigning values to $x_1$, $x_2$, and $x_3$ based on their organization scores:
- $x_1 = 5.9 / 5.9 = 1$ (Implicitly, if we consider the score as a direct value)
- $x_2 = 19.39 / 19.39 = 1$
- $x_3 = 20.41 / 20.41 = 1$

But given the problem statement, it seems more accurate that $x_1$, $x_2$, and $x_3$ represent hours worked and their organization scores are given as:
- $r_0 = 5.9x_1 + 19.39x_2 + 20.41x_3$ with an upper bound of 306, and specific scores for each.

## 5: Correctly interpret the constraints
Given:
- The organization score for Hank is 5.9
- The organization score for Jean is 19.39
- The organization score for Mary is 20.41

And the constraints:
- $5.9x_1 + 20.41x_3 \geq 87$
- $5.9x_1 + 19.39x_2 \geq 61$
- $5.9x_1 + 19.39x_2 + 20.41x_3 \geq 61$
- $-10x_1 + 6x_3 \geq 0$
- $7x_2 - 9x_3 \geq 0$
- $5.9x_1 + 19.39x_2 \leq 175$

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Jean'), ('x3', 'hours worked by Mary')],
    'objective_function': '6.35x1 + 4.82x2 + 7.75x3',
    'constraints': [
        '5.9x1 + 20.41x3 >= 87',
        '5.9x1 + 19.39x2 >= 61',
        '5.9x1 + 19.39x2 + 20.41x3 >= 61',
        '-10x1 + 6x3 >= 0',
        '7x2 - 9x3 >= 0',
        '5.9x1 + 19.39x2 <= 175'
    ]
}
```

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

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

    # Define the variables
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Hank
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Jean
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Mary

    # Define the objective function
    model.setObjective(6.35 * x1 + 4.82 * x2 + 7.75 * x3, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(5.9 * x1 + 20.41 * x3 >= 87)
    model.addConstr(5.9 * x1 + 19.39 * x2 >= 61)
    model.addConstr(5.9 * x1 + 19.39 * x2 + 20.41 * x3 >= 61)
    model.addConstr(-10 * x1 + 6 * x3 >= 0)
    model.addConstr(7 * x2 - 9 * x3 >= 0)
    model.addConstr(5.9 * x1 + 19.39 * x2 <= 175)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Hank: ", x1.varValue)
        print("Hours worked by Jean: ", x2.varValue)
        print("Hours worked by Mary: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```