Practice GMAT Data Sufficiency Question

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Is (2y+z)(3x)(5y)(7z) < (90y)(14z)?
  1. y and z are positive integers; x = 1
  2. x and z are positive integers; y = 1
Correct Answer: A
  1. Simplify the equation:
    Is (2y+z)(3x)(5y)(7z) < (90y)(14z)?
    Simplified: is (2y+z)(3x)(5y)(7z) < ((2*5*3*3)y)((7*2)z)?
    Simplified: is (2y+z)(3x)(5y)(7z) < (2y)(5y)(32y)(7z)(2z)?
    Simplified: is (2y+z)(3x)(5y)(7z) < (2y+z)(5y)(32y)(7z)?
    Cancel out 2y+z, 5y, and 7z
    Simplified: is 3x < 32y?
  2. Evaluate Statement (1) alone.
    1. Statement (1) says that x = 1. So, plug that information in and work from there.
      Simplified Question: is 31 < 32y 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, 3x will always be less than 32y.
      Statement (1) is SUFFICIENT.
  3. 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 3x < 32(1)?
      Or: is 3x < 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.
  4. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

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