A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff...

Understanding the Question

What we need to find: The number of staff members in the department.

Given information:

• Each staff member received the same quantities: \(\mathrm{x}\) pens, \(\mathrm{y}\) pencils, and \(\mathrm{z}\) pads
• We need to determine exactly one value for the number of staff members

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.

Analyzing Statement 1

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:

• The actual number of items each person received
• The total number of items distributed
• The number of staff members

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.

Analyzing Statement 2

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

• Number of staff = \(18 \div 2 = 9\) people ✓
• Check: \(27 \div 3 = 9\) people ✓ and \(36 \div 4 = 9\) people ✓
• All items distributed perfectly!

Scenario 2: Each person gets 6 pens, 9 pencils, 12 pads

• Number of staff = \(18 \div 6 = 3\) people ✓
• Check: \(27 \div 9 = 3\) people ✓ and \(36 \div 12 = 3\) people ✓
• All items distributed perfectly!

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.

Combining Both Statements

Now let's use both statements together to see if we can determine exactly one answer.

From both statements we know:

• Each person gets items in ratio \(2:3:4\) (Statement 1)
• Total items are 18 pens, 27 pencils, 36 pads (Statement 2)

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

• Uses \(9 \times 2 = 18\) pens, \(9 \times 3 = 27\) pencils, \(9 \times 4 = 36\) pads ✓
• Each person's items are in ratio \(2:3:4\) ✓

Valid distribution 2: 3 people each getting \((6,9,12)\) items

• Uses \(3 \times 6 = 18\) pens, \(3 \times 9 = 27\) pencils, \(3 \times 12 = 36\) pads ✓
• Each person's items are in ratio \(6:9:12 = 2:3:4\) ✓

Valid distribution 3: 1 person getting \((18,27,36)\) items

• Uses all items ✓
• Items are in ratio \(18:27:36 = 2:3:4\) ✓

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 Answer: E

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."