Work and Rate Problems

What we see: A Data Sufficiency work-rate question. We need one exact number: the fewest identical pipes that fill the tank within 30 minutes.

Step 1. Know what we need.
What we see: all the pipes are identical.
What it means: if we know how long one pipe takes, we can work out any number of pipes.
What we do and why: call one pipe's time \(P\) minutes. Then \(n\) pipes take \(\frac{P}{n}\) minutes, so we need the smallest \(n\) with \(\frac{P}{n} \le 30\). The only thing we need is \(P\).

Step 2. Statement 1 alone.
What we see: 8 pipes finish 20 minutes sooner than 6 pipes.
What it means: \(\frac{P}{6} - \frac{P}{8} = 20\).
What we do and why: \(\frac{4P - 3P}{24} = 20\), so \(\frac{P}{24} = 20\) and \(P = 480\). Then \(\frac{480}{n} \le 30\) gives \(n \ge 16\). One answer: 16 pipes. Sufficient.

Step 3. Statement 2 alone.
What we see: 20 pipes take less than 15 minutes.
What it means: \(\frac{P}{20} < 15\), so \(P < 300\). \(P\) could be 290, or it could be 10.
What we do and why: we test two values. If \(P = 290\), we need 10 pipes. If \(P = 10\), we need 1 pipe. Two different answers. Not sufficient.

The trap: thinking Statement 2 is enough because it also talks about filling time. It only says one pipe takes under 300 minutes, a range, so the number of pipes is not fixed.
Faster way: we only need \(P\). Statement 1 is one equation with \(P\) as its only unknown, so it is sufficient before you solve it.
Takeaway: in Data Sufficiency, one equation in one unknown is enough; an inequality usually leaves a range and is not.

Answer: A