The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

Write x for the number of hats Xander has, y for the number of hats Yolanda has, and z for the number of hats Zelda has. From the question stem, we know that x < y < z and that x + y + z = 12. Moreover, since each person has at least one hat, and people can only have integer numbers of hats, we know that x, y, and z are all positive integers. With this number of constraints, we should go ahead and list scenarios that fit all the constraints. Start with x and y as low as possible, then adjust from there, keeping the order, keeping the sum at 12, and ensuring that no two integers are the same.

Scenario x y z
(a) 1 2 9
(b) 1 3 8
(c) 1 4 7
(d) 1 5 6
(e) 2 3 7
(f) 2 4 6
(g) 3 4 5

These are the only seven scenarios that work. As a reminder, we are looking for the value of y. Now, we turn to the statements.

Statement (1): INSUFFICIENT. We are told that zx is less than or equal to 5. This rules out scenarios (a) through (c), but the last four scenarios still work. Thus, y could be 3, 4, or 5.

Statement (2): INSUFFICIENT. We are told that xyz is less than 36. We work out this product for the seven scenarios:

(a) 18

(b) 24

(c) 28

(d) 30

(e) 42

(f) 48

(g) 60

We can rule out scenarios (e) through (g), but (a) through (d) still work. Thus, y could be 2, 3, 4, or 5.

Statements (1) and (2) together: SUFFICIENT. Only scenario (d) survives the constraints of the two statements. Thus, we know that y is 5.

The correct answer is (C): BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient.

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