In the next month, Technology Firm X will hire exactly 24 new engineers to at least partially fulfill the hiring...
GMAT Two Part Analysis : (TPA) Questions
In the next month, Technology Firm X will hire exactly \(24\) new engineers to at least partially fulfill the hiring needs of the firm's \(3\) engineering departments: the mechanical design department, which needs \(15\) new engineers; the electrical design department, which needs \(12\) new engineers; and the software design department, which needs \(8\) new engineers. Each of the engineering departments will receive at least \(1\) of the \(24\) new engineers, while none of the departments will receive more than it needs. Each of the new engineers will be employed in exactly one of the three departments. Assume that in every case in which a department \(\mathrm{D_1}\) needs more engineers than a department \(\mathrm{D_2}\), \(\mathrm{D_1}\) will receive more engineers than \(\mathrm{D_2}\).
Select the minimum number of engineers that the mechanical design department could receive, given this assumption and the information provided. And select the minimum number of engineers that the electrical design department could receive, given this assumption and the information provided. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visualization
Let's create a table to organize our information:
| Department | Engineers Needed | Engineers Received | Constraints |
| Mechanical | 15 | M | \(\mathrm{M} \geq 1, \mathrm{M} \leq 15\) |
| Electrical | 12 | E | \(\mathrm{E} \geq 1, \mathrm{E} \leq 12\) |
| Software | 8 | S | \(\mathrm{S} \geq 1, \mathrm{S} \leq 8\) |
Key relationships:
- Total: \(\mathrm{M} + \mathrm{E} + \mathrm{S} = 24\)
- Ordering by needs: Mechanical (15) > Electrical (12) > Software (8)
- Critical constraint: Since departments that need more must receive more:
- \(\mathrm{M} > \mathrm{E}\) (Mechanical receives more than Electrical)
- \(\mathrm{M} > \mathrm{S}\) (Mechanical receives more than Software)
- \(\mathrm{E} > \mathrm{S}\) (Electrical receives more than Software)
- Combined: \(\mathrm{M} > \mathrm{E} > \mathrm{S}\)
Phase 2: Understanding the Question
We need to find:
- Minimum number of engineers Mechanical could receive
- Minimum number of engineers Electrical could receive
These are separate optimization problems with the same constraints.
Phase 3: Finding the Answer
Finding Minimum for Mechanical (M)
To minimize M, we need to maximize E and S while maintaining \(\mathrm{M} > \mathrm{E} > \mathrm{S}\).
Since \(\mathrm{S} \leq 8\) and \(\mathrm{E} > \mathrm{S}\), let's work backwards:
- If \(\mathrm{S} = 7\), then E must be at least 8
- If \(\mathrm{E} = 8\), then M must be at least 9
- Check: \(\mathrm{M} + \mathrm{E} + \mathrm{S} = 9 + 8 + 7 = 24\) ✓
- Verify constraints: \(9 > 8 > 7\) ✓, and \(9 \leq 15, 8 \leq 12, 7 \leq 8\) ✓
Can we make M smaller? If \(\mathrm{S} = 8\):
- E must be at least 9 (since \(\mathrm{E} > \mathrm{S}\))
- M must be at least 10 (since \(\mathrm{M} > \mathrm{E}\))
- Sum would be at least \(10 + 9 + 8 = 27 > 24\) ✗
Minimum M = 9
Finding Minimum for Electrical (E)
To minimize E, we need to maximize M and minimize S while maintaining \(\mathrm{M} > \mathrm{E} > \mathrm{S}\).
Let's maximize \(\mathrm{M} = 15\) and see how small E can be:
- We need \(15 + \mathrm{E} + \mathrm{S} = 24\), so \(\mathrm{E} + \mathrm{S} = 9\)
- Since \(\mathrm{E} > \mathrm{S}\) and \(\mathrm{S} \geq 1\), let's try different values:
If \(\mathrm{S} = 4\):
- \(\mathrm{E} = 9 - 4 = 5\)
- Check: \(15 > 5 > 4\) ✓
- All constraints satisfied ✓
Can S be larger? If \(\mathrm{S} = 5\):
- \(\mathrm{E} = 9 - 5 = 4\)
- But then \(\mathrm{E} > \mathrm{S}\) means \(4 > 5\), which is false ✗
Minimum E = 5
Phase 4: Solution
Final Answer:
- Minimum for Mechanical Design Department: 9
- Minimum for Electrical Design Department: 5
These values satisfy all constraints:
- When Mechanical gets minimum (9): Distribution is \(\mathrm{M}=9, \mathrm{E}=8, \mathrm{S}=7\)
- When Electrical gets minimum (5): Distribution is \(\mathrm{M}=15, \mathrm{E}=5, \mathrm{S}=4\)
- Both maintain the required ordering \(\mathrm{M} > \mathrm{E} > \mathrm{S}\) based on departmental needs