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A bank has 2 paper shedding machines, one large and one small, to shred sheets of secure waste paper, all of which are the same size. The large machine shreds sheets at a constant rate that is 5 times the constant rate of the small machine. How many minutes would it take the small machine, working alone at its constant shredding rate, to shred \(\mathrm{n}\) sheets of the paper?
We need to find: How many minutes would it take the small machine, working alone at its constant shredding rate, to shred n sheets of paper?
What we know:
This asks for a specific value - an exact number of minutes. We need enough information to calculate that precise time.
When machines work together, their contributions are proportional to their speeds. Since the large machine works 5 times as fast, in any time period:
Think of it like two workers where one is 5 times faster - the faster worker will complete 5 parts while the slower completes 1 part.
Statement 1: Working together at their individual constant rates, the machines could shred n sheets in 40 minutes.
Let's reason through this step by step:
When the machines work together for 40 minutes:
Since the small machine completes \(\frac{n}{6}\) sheets in 40 minutes, how long would it need for all n sheets?
If you complete \(\frac{1}{6}\) of a job in 40 minutes, you need 6 times as long for the whole job:
Time = \(6 \times 40 = 240\) minutes
We can determine the exact time: 240 minutes.
[STOP - Sufficient!] Statement 1 alone is sufficient.
This eliminates choices B, C, and E.
Forget Statement 1 completely. Look only at Statement 2.
Statement 2: \(n = 1,200\)
This tells us the quantity of sheets, but nothing about the machines' speeds.
Think of it this way: If I tell you that you need to shred 1,200 sheets, can you tell me how long it will take? No - because you don't know how fast you can shred.
Without knowing:
We cannot determine how long it takes the small machine to shred 1,200 sheets.
Statement 2 alone is NOT sufficient.
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."