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 z – x 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.