Machine M, working alone at its constant rate, produces x widgets every 4 minutes. Machine N, working alone at its...

Understanding the Question

Let's understand what we're comparing here. Machine M produces x widgets every 4 minutes, while Machine N produces y widgets every 5 minutes. Both work for 20 minutes.

What We Need to Determine

In 20 minutes:

• Machine M completes \(20 \div 4 = 5\) production cycles
• Machine N completes \(20 \div 5 = 4\) production cycles

Therefore, Machine M produces 5x widgets while Machine N produces 4y widgets.

The question asks: Is \(5\mathrm{x} > 4\mathrm{y}\)?

This is a yes/no question. We need to determine definitively whether Machine M produces more widgets than Machine N.

Key Insight

Notice that Machine M has a built-in advantage - it gets 25% more production cycles (5 vs 4). The question essentially asks whether this cycle advantage is enough to overcome any potential disadvantage in per-cycle production.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{x} > 0.8\mathrm{y}\)

This means Machine M produces more than 80% of what Machine N produces per cycle.

Testing the Relationship

Let's think about this strategically. Machine M already gets 5 cycles while N gets only 4. Now we learn that M produces more than 80% of N's per-cycle output.

Even in the worst-case scenario where x is barely above 0.8y:

• If N produces 100 widgets per cycle, M produces just over 80
• In 20 minutes: M produces \(5 \times 80+ = 400+\) widgets
• While N produces \(4 \times 100 = 400\) widgets

Since M always produces more than 80% per cycle AND gets 25% more cycles, Machine M will always produce more total widgets.

[STOP - Sufficient!]

Conclusion

Statement 1 is sufficient to answer YES - Machine M produces more widgets than Machine N.

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(\mathrm{y} = \mathrm{x} + 1\)

This means Machine N produces exactly 1 more widget per cycle than Machine M.

Testing Different Scenarios

The question now becomes: Can M's extra cycle overcome N's advantage of 1 extra widget per cycle? Let's test concrete scenarios:

Small production scenario:

• If \(\mathrm{x} = 2\), then \(\mathrm{y} = 3\)
• M produces: \(5 \times 2 = 10\) widgets
• N produces: \(4 \times 3 = 12\) widgets
• Answer: NO, M does not produce more

Large production scenario:

• If \(\mathrm{x} = 10\), then \(\mathrm{y} = 11\)
• M produces: \(5 \times 10 = 50\) widgets
• N produces: \(4 \times 11 = 44\) widgets
• Answer: YES, M produces more

Different values of x lead to different answers to our question.

Why Different Outcomes?

The key is that N's advantage is fixed (always 1 widget per cycle), while M's cycle advantage becomes more powerful with larger production numbers. For small x values, N's fixed advantage wins. For large x values, M's percentage-based cycle advantage wins.

Conclusion

Statement 2 is NOT sufficient because we get different answers depending on the actual values.

This eliminates choices B and D.

The Answer: A

Since Statement 1 alone is sufficient but Statement 2 alone is not sufficient, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."