Machines K, M, and N, each working alone at its constant rate, produce 1 widget in x, y, and 2...

Understanding the Question

We need to determine: Do three machines working together produce 50 widgets in less than 60 minutes?

Given Information

• Machine K produces 1 widget in x minutes (rate = \(\frac{1}{x}\) widgets per minute)
• Machine M produces 1 widget in y minutes (rate = \(\frac{1}{y}\) widgets per minute)
• Machine N produces 1 widget in 2 minutes (rate = \(\frac{1}{2}\) widgets per minute)
• All machines work simultaneously at constant rates

What We Need to Find

We need a definitive YES or NO answer to whether their combined production time for 50 widgets is less than 60 minutes.

Key Insight

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."

Analyzing Statement 1

Statement 1: x < 1.5

This tells us Machine K produces each widget in less than 1.5 minutes.

The Logical Path

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):

• Combined rate > \(\frac{2}{3} + 0 + \frac{1}{2} = \frac{7}{6}\) widgets per minute

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.

Analyzing Statement 2

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.

The Revealing Insight

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:

• Combined rate > \(0 + \frac{5}{6} + \frac{1}{2} = \frac{8}{6} = \frac{4}{3}\) widgets per minute

Since \(\frac{4}{3} > \frac{5}{6}\), the answer is definitively YES.

[STOP - Statement 2 is Sufficient!]

The Answer: D

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.