Each of the 75 employees at a certain company works in exactly one of the company's 3 departments (Departments X,...

Phase 1: Owning the Dataset

Visualization Selection

We have a problem about employees distributed across departments. This is best represented with a visual breakdown table showing how the total is partitioned.

Visual Representation

Total Employees: 75

Department X: [***] (20% of 75)
Department Y: [?????] (10 fewer than Z)
Department Z: [???????] (unknown)

Given Information

• Total employees: 75
• Exactly 3 departments: X, Y, and Z
• Each employee works in exactly one department
• Department X: 20% of employees
• Department Y: 10 fewer employees than Department Z

Initial Calculations

Department X has: \(20\% \times 75 = 0.20 \times 75 = \mathbf{15}\) employees

Phase 2: Understanding the Question

We need to find the exact number of employees in:

• Department Y (first selection)
• Department Z (second selection)

Setting Up the Mathematical Relationships

Let's denote:

• Department Y has y employees
• Department Z has z employees

From our given information:

1. \(\mathrm{y} = \mathrm{z} - 10\) (Y has 10 fewer than Z)
2. \(15 + \mathrm{y} + \mathrm{z} = 75\) (all employees must sum to 75)

Simplifying equation 2:
\(\mathrm{y} + \mathrm{z} = 60\)

Phase 3: Finding the Answer

Solving the System

We have:

• \(\mathrm{y} = \mathrm{z} - 10\)
• \(\mathrm{y} + \mathrm{z} = 60\)

Substituting the first equation into the second:
\((\mathrm{z} - 10) + \mathrm{z} = 60\)
\(2\mathrm{z} - 10 = 60\)
\(2\mathrm{z} = 70\)
\(\mathbf{\mathrm{z} = 35}\)

Therefore:
\(\mathbf{\mathrm{y} = \mathrm{z} - 10 = 35 - 10 = 25}\)

Verification

Let's check our answer:

• Department X: 15 employees
• Department Y: 25 employees
• Department Z: 35 employees
• Total: \(15 + 25 + 35 = 75\) ✓

Also verifying the constraint: Y has 10 fewer than Z: \(25 = 35 - 10\) ✓

Updated Visual

Total Employees: 75

Department X: [***] 15 employees (20%)
Department Y: [*****] 25 employees
Department Z: [*******] 35 employees

Phase 4: Solution

Final Answer:

• Department Y: 25
• Department Z: 35

These values satisfy all our constraints: they sum to 75 total employees (with X's 15), and Department Y has exactly 10 fewer employees than Department Z.