Machine M and Machine N working alone at their constant rates, non stop, produced 6000 and 8000 nails respectively. Did...

Understanding the Question

Let's understand what we're asking: Did machine M work longer than machine N?

Given Information

• Machine M produced 6,000 nails total
• Machine N produced 8,000 nails total
• Both machines work at constant rates

What We Need to Determine

We need to know if Machine M's working time > Machine N's working time.

Here's the key relationship: Working time = Total nails produced ÷ Production rate

So we're asking: Is \(\frac{6000}{\mathrm{M's\ rate}} > \frac{8000}{\mathrm{N's\ rate}}\)?

This is a yes/no question - we need to determine with certainty whether M worked longer or not.

Key Insight

Machine M made fewer nails (6,000 vs 8,000). This might suggest it worked less time. BUT if Machine M is significantly slower than N, it could still work longer despite making fewer nails. The answer depends entirely on how their production rates compare.

Analyzing Statement 1

Statement 1 tells us: Machine N produces 2,000 more nails than Machine M in one hour.

This gives us a fixed difference between their rates: \(\mathrm{N's\ rate} = \mathrm{M's\ rate} + 2000\) nails/hour.

Testing Different Scenarios

Let's test extreme cases to see if we always get the same answer:

Scenario 1: Very slow rates

• If M produces 100 nails/hour, then N produces 2,100 nails/hour
• M's time: \(6000 \div 100 = 60\) hours
• N's time: \(8000 \div 2100 \approx 3.8\) hours
• Result: M works much longer than N ✓

Scenario 2: Very fast rates

• If M produces 10,000 nails/hour, then N produces 12,000 nails/hour
• M's time: \(6000 \div 10000 = 0.6\) hours
• N's time: \(8000 \div 12000 \approx 0.67\) hours
• Result: N works longer than M ✗

Since different scenarios give different answers to our yes/no question, we cannot determine with certainty whether M worked longer than N.

Conclusion

Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2 tells us: Machine N produces twice as much as Machine M in one hour.

This gives us a ratio: \(\mathrm{N's\ rate} = 2 \times \mathrm{M's\ rate}\).

Logical Analysis

Here's the elegant insight: If N is twice as fast as M, but N only made \(\frac{8000}{6000} = 1.33\times\) as many total nails, then N must have worked for less time than M.

Let's think about it this way:

• If M works for some time T and makes 6,000 nails
• At double the rate, N would make 12,000 nails in the same time T
• But N only made 8,000 nails (not 12,000)
• Therefore, N must have stopped working before time T

Quick Verification

Let's verify with any rate for M:

• If M's rate = 1,000 nails/hour, then N's rate = 2,000 nails/hour
• M's time: \(6000 \div 1000 = 6\) hours
• N's time: \(8000 \div 2000 = 4\) hours
• Result: M works longer ✓

No matter what M's actual rate is, this relationship always holds true: M worked longer than N.

[STOP - Sufficient!]

Conclusion

Statement 2 is sufficient to answer that yes, Machine M worked longer than Machine N.

This eliminates choices C and E.

The Answer: B

Statement 2 alone tells us definitively that Machine M worked longer than Machine N, while Statement 1 alone does not provide enough information.

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