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A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff member receiving \(\mathrm{x}\) pens, \(\mathrm{y}\) pencils, and \(\mathrm{z}\) pads. How many staff members were in the department?
What we need to find: The number of staff members in the department.
Given information:
What would be sufficient: We need enough information to determine exactly one possible value for the number of staff members. If multiple values are possible, the information is not sufficient.
Key insight: Since total items = (number of people) × (items per person), we need information that uniquely determines this relationship.
Statement 1 tells us: The items each person received are in the ratio \(2:3:4\) (pens:pencils:pads).
This means if someone gets 2 pens, they get 3 pencils and 4 pads. Or if they get 4 pens, they get 6 pencils and 8 pads. The ratio stays constant, but the actual quantities can vary.
What we still don't know:
Think of it this way: The ratio is like knowing recipe proportions without the serving size. Whether each person gets \((2,3,4)\) or \((20,30,40)\) or \((200,300,400)\) items, the ratio holds. Without knowing either the total items or the actual per-person amounts, we can't determine how many people there are.
[STOP - Not Sufficient!]
Conclusion: Statement 1 is NOT sufficient.
This eliminates answer choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The manager distributed totals of 18 pens, 27 pencils, and 36 pads.
What we still don't know: How many items each person received (the values of \(\mathrm{x}\), \(\mathrm{y}\), and \(\mathrm{z}\)).
Let's test different scenarios to see if we get a unique answer:
Scenario 1: Each person gets 2 pens, 3 pencils, 4 pads
Scenario 2: Each person gets 6 pens, 9 pencils, 12 pads
Both scenarios use up all the items exactly, but give different numbers of staff members (9 vs 3). Since we found two different valid answers, Statement 2 alone doesn't determine a unique answer.
[STOP - Not Sufficient!]
Conclusion: Statement 2 is NOT sufficient.
This eliminates answer choice B.
Now let's use both statements together to see if we can determine exactly one answer.
From both statements we know:
Key observation: Notice that the totals \((18:27:36)\) can be simplified to the same ratio \(2:3:4\). This is important because it means multiple equal distributions are possible while satisfying both conditions!
Let's verify the possible distributions:
Valid distribution 1: 9 people each getting \((2,3,4)\) items
Valid distribution 2: 3 people each getting \((6,9,12)\) items
Valid distribution 3: 1 person getting \((18,27,36)\) items
Since we have three different valid answers (1, 3, or 9 staff members), even both statements together don't give us a unique answer.
[STOP - Not Sufficient!]
Conclusion: The statements together are NOT sufficient.
The statements together are not sufficient because multiple values (1, 3, or 9) for the number of staff members satisfy both conditions.
Answer Choice E: "The statements together are not sufficient."