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Machine M, working alone at its constant rate, produces \(\mathrm{x}\) widgets every 4 minutes. Machine N, working alone at its constant rate, produces \(\mathrm{y}\) widgets every 5 minutes. If machines M and N working simultaneously at their respective constant rates for 20 minutes, does machine M produce more widgets than machine N in that time ?
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.
In 20 minutes:
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.
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.
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.
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:
Since M always produces more than 80% per cycle AND gets 25% more cycles, Machine M will always produce more total widgets.
[STOP - Sufficient!]
Statement 1 is sufficient to answer YES - Machine M produces more widgets than Machine N.
This eliminates choices B, C, and E.
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.
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:
Large production scenario:
Different values of x lead to different answers to our question.
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.
Statement 2 is NOT sufficient because we get different answers depending on the actual values.
This eliminates choices B and D.
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."