GMAT Focus EditionQuantitative ReasoningHard
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 operates continuously. If the machine produces 50 defective items when operating for 5 hours, how many defective items will it produce if it operates for 8 hours continuously?
- A200
- B128
- C80
- D160
Show answer & explanationAnswer & explanation
Correct answer: B. 128
The relationship is Defective Items (D) = k * (Hours)^2. Using the initial condition, 50 = k * (5)^2, so k = 50/25 = 2. Then, for 8 hours, D = 2 * (8)^2 = 2 * 64 = 128 defective items.
Why the other options are wrong
- A. Incorrect calculation based on the square relationship.
- C. This would be the answer if it was directly proportional to hours, not square of hours (50/5 = 10; 10*8 = 80).
- D. Incorrect calculation based on the square relationship.
Direct Proportion (with powers)
Two quantities are in direct proportion with a power if one quantity varies directly as a power (square, cube, etc.) of the other, meaning their ratio is a constant (y = kx^n).
- Expressed as y = kx^n, where k is the constant of proportionality.
- As x increases, y increases (or decreases if n is negative).
- Finding 'k' is crucial to solve for unknown values.
Memory trick: Constant K Links Powerful Proportions.