Manhattan GMAT Challenge Problem of the Week – 15 January 2013
Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!
If = 25, = 64, and = 216, and xy > 0, which of the following is true?
A. x > y > z
B. y > x > z
C. y > z > x
D. z > y > x
E. z > x > y
First, rewrite the right sides of the given equations as powers of small integers:
= 25 =
= 64 =
= 216 = (you might not have recognized this one, but you could get there with prime factorization)
Take roots as necessary to isolate each variable. You know that x and z must be positive, because their odd powers are positive; moreover, since xy > 0, y must also be positive.
y = =
Now compare the results in pairs, putting “??” between to indicate the unknown inequality. First, compare x and y:
Raise both sides to the 6th power, to remove the fractional powers:
625 ?? 512
Since 625 is bigger than 512, x is bigger than y. You can now eliminate B, C, and D.
The remaining two answers differ as to whether z is largest or smallest. Since y seems simpler than x (both in the base and in the exponent), compare y and z:
Raise both sides to the 10th power, to remove the fractional powers:
512 ?? = = 81 × 9 = 729
So z is larger than y, eliminating A. The answer must therefore be E.
Incidentally, it’s very difficult to prove by hand that z is bigger than x. To do so, you need to show that is bigger than , very much a non-trivial task! Fortunately, you don’t need to determine whether z or x is larger in order to get the right answer.
The correct answer is E.
Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.