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GMAT Focus EditionQuantitative ReasoningMedium

A production manager observes that the number of defective items produced by a machine is directly proportional to the square of the number of hours the machine has been running. If the machine produces 12 defective items in 2 hours, how many defective items will it produce in 5 hours?

  1. A60
  2. B30
  3. C150
  4. D75
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Correct answer: D. 75

Let 'D' be the number of defective items and 'H' be the number of hours the machine runs. The problem states D is directly proportional to the square of H, so D = kH^2, where 'k' is the constant of proportionality. Given D = 12 when H = 2: 12 = k(2^2) => 12 = 4k => k = 3. Now we have the relationship D = 3H^2. We need to find D when H = 5: D = 3(5^2) = 3(25) = 75. Thus, the machine will produce 75 defective items in 5 hours.

Why the other options are wrong

  • A. Incorrect. This is double the linear proportion, or 3 * (5*4) = 60, not using the square correctly.
  • B. Incorrect. This would be if it were directly proportional to H (12/2 * 5 = 30).
  • C. Incorrect. This would be 3 * (5^2 * 2) if an extra factor of 2 was incorrectly included.

Direct Proportion (with powers)

Direct proportion with powers describes a relationship where one quantity varies directly as a power of another quantity. If y is directly proportional to x^n, then y = kx^n for some constant k.

  • The relationship can be written as y = kx^n.
  • First, use given values to find the constant of proportionality 'k'.
  • Then, use 'k' and the new value of 'x' to find the new 'y'.
  • Common powers include square (n=2) or cube (n=3).

Memory trick: Constant 'k' is the key; power 'n' rules the variety.

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