Each of the 75 employees at a certain company works in exactly one of the company's 3 departments (Departments X,...
GMAT Two Part Analysis : (TPA) Questions
Each of the 75 employees at a certain company works in exactly one of the company's 3 departments (Departments X, Y, and Z). Exactly \(20\%\) of the employees work in Department X, and 10 fewer employees work in Department Y than work in Department Z.
Based on the information provided, select for Department Y and Department Z the numbers of employees who work in Department Y and Department Z. Make only two selections, one in each column.
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:
- \(\mathrm{y} = \mathrm{z} - 10\) (Y has 10 fewer than Z)
- \(15 + \mathrm{y} + \mathrm{z} = 75\) (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.