Practice GMAT Data Sufficiency Question

Return to the list of practice GMAT data sufficiency questions.

Is (2^y+z)(3^x)(5^y)(7^z) < (90^y)(14^z)?

y and z are positive integers; x = 1
x and z are positive integers; y = 1

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)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: A

Simplify the equation:
Is (2^y+z)(3^x)(5^y)(7^z) < (90^y)(14^z)?
Simplified: is (2^y+z)(3^x)(5^y)(7^z) < ((2*5*3*3)^y)((7*2)^z)?
Simplified: is (2^y+z)(3^x)(5^y)(7^z) < (2^y)(5^y)(3^2y)(7^z)(2^z)?
Simplified: is (2^y+z)(3^x)(5^y)(7^z) < (2^y+z)(5^y)(3^2y)(7^z)?
Cancel out 2^y+z, 5^y, and 7^z
Simplified: is 3^x < 3^2y?
Evaluate Statement (1) alone.
1. Statement (1) says that x = 1. So, plug that information in and work from there.
  Simplified Question: is 3¹ < 3^2y where y is a positive integer?
  Further Simplified: is 1 < 2y where y is a positive integer?
2. At this point, some students can see that Statement (1) is SUFFICIENT. However, a more thorough analysis is provided just to be clear.
3. Since x and y are given as positive integers, the smallest possible value for y is 1. In this case 1 < 2(1). Since the inequality held true when y=1, it will hold true for any legal value of y since y will only get larger and x will not change.
4. Thus, 3^x will always be less than 3^2y.
  Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. Statement (2) says y = 1 and x and z are positive integers. So, plug that information and work from there.
  Is 3^x < 3²⁽¹⁾?
  Or: is 3^x < 3²?
  Or: is x < 2?
2. Since the only restriction on x is that it is a positive integer, x could be 1 (in which case the inequality would be true and the answer to the question would be "Yes") or, x could be 2 (in which case the inequality would not be true and the answer to the question would be "No").
3. Since different answers to the question "is x < 2?" are possible, there is no definitive answer to the question. Statement (2) is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

Return to the list of practice GMAT data sufficiency questions.