Question

A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff member receiving $x$ pens, $y$ pencils, and $z$ pads. How many staff members were in the department?
(1) The numbers of pens, pencils, and pads that each staff member received were in the ratio $2:3:4$, respectively.
(2) The manager distributed a total of $18$ pens, $27$ pencils, and $36$ pads.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is E Statements (1) and (2) together are not sufficient.

1. Considering statement 1, each of 10 staff members could have received 2 pens, 3 pencils, and 4 pads, or each of 20 staff members could have received 2 pens, 3 pencils, and 4 pads; NOT sufficient.
2. Considering statement 2, there could have been 1 staff member who received 18 pens, 27 pencils, and 36 pads, or 3 staff members each of whom received 6 pens, 9 pencils, and 12 pads; NOT sufficient.

• Assuming both (1) and (2), use the fact that 18 : 27 : 36 is equivalent to both 6 : 9 : 12 and 2 : 3 : 4 to obtain different possibilities for the number of staff. Each of 3 staff members could have received 6 pens, 9 pencils, and 12 pads, or each of 9 staff members could have received 2 pens, 3 pencils, and 4 pads.
• Therefore, (1) and (2) together are not sufficient. The correct answer is E; both statements together are still not sufficient.