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Machines K, M, and N, each working alone at its constant rate, produce \(1\) widget in \(\mathrm{x}\), \(\mathrm{y}\), and \(2\) minutes, respectively. If Machines K, M, and N work simultaneously at their respective constant rates, does it take them less than \(1\) hour to produce a total of \(50\) widgets?
We need to determine: Do three machines working together produce 50 widgets in less than 60 minutes?
We need a definitive YES or NO answer to whether their combined production time for 50 widgets is less than 60 minutes.
To produce 50 widgets in less than 60 minutes, the machines must produce MORE than \(\frac{50}{60} = \frac{5}{6}\) widget per minute combined. Since Machine N contributes exactly \(\frac{1}{2}\) widget per minute, Machines K and M together must contribute more than \(\frac{5}{6} - \frac{1}{2} = \frac{1}{3}\) widget per minute.
Translation: The question is really asking whether Machines K and M are fast enough to "make up the difference."
Statement 1: x < 1.5
This tells us Machine K produces each widget in less than 1.5 minutes.
If Machine K takes less than 1.5 minutes per widget, it produces MORE than \(\frac{1}{1.5} = \frac{2}{3}\) widget per minute.
But remember: we only need K and M together to produce more than \(\frac{1}{3}\) widget per minute. Machine K alone already produces more than \(\frac{2}{3}\) widget per minute—that's already double what we need from both K and M combined!
Even in the worst case (if Machine M produces nothing):
Since \(\frac{7}{6} > \frac{5}{6}\), the answer is definitively YES.
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E. We're down to A or D.
Statement 2: y < 1.2
Remember: We must analyze this independently, forgetting Statement 1 completely.
This tells us Machine M produces each widget in less than 1.2 minutes.
Here's a quick mental calculation: If Machine M takes exactly 1.2 minutes per widget, that's 5 widgets in 6 minutes, which equals \(\frac{5}{6}\) widget per minute. Since y < 1.2, Machine M produces MORE than \(\frac{5}{6}\) widget per minute.
But wait—we only needed \(\frac{5}{6}\) widget per minute from ALL THREE machines combined! Machine M alone already exceeds our total requirement.
Even if Machine K produces nothing:
Since \(\frac{4}{3} > \frac{5}{6}\), the answer is definitively YES.
[STOP - Statement 2 is Sufficient!]
Both statements independently guarantee that the machines will produce 50 widgets in less than 60 minutes. The key insight: each statement tells us that one of the variable-speed machines is so fast that it single-handedly ensures we exceed the required production rate.
Answer Choice D: Each statement alone is sufficient.