A few posts ago, I tackled rate/work questions, which are invariably a source of consternation for GMAT test-takers. On the latest official practice tests that GMAC has released, these questions showed up with surprising frequency, so I thought it might be worthwhile to tackle a challenging incarnation of this question type: one in which a single machine begins a project and then multiple machines complete the partially-finished work.
To review, the key for dealing with this type of question is to apply the following rules:
- Rate * Time = Work
- Rates are additive in work questions.
- Rate and time have a reciprocal relationship.
For the questions involving partially completed jobs, we’ll throw in the addendum that a completed job can be designated as “1”’
And that’s it!
Here’s a question I saw on my recent practice test:
Working alone at its constant rate, pump X pumped out ¼ of the water in a tank in 2 hours. Then pumps Y and Z started working and the three pumps, working simultaneously at their respective constant rates, pumped out the rest of the water in 3 hours. If pump Y, working alone at its constant rate, would have taken 18 hours to pump out the rest of the water, how many hours would it have taken pump Z, working alone at its constant rate, to pump out all of the water that was pumped out of the tank?
Okay, deep breath. Recall our three aforementioned rules. Next, let’s designate the rates for the pumps as x, y, and z, respectively.
If pump x can pump out ¼ of the water in 2 hours, then it would take 4*2 = 8 hours to pump out all the water alone. If pump x can complete 1 tank in 8 hours, then x = 1/8.
If x removes ¼ of the water on its own, then all three pumps working together have to remove the ¾ of the water left in the tank. We’re told that together they can do this in 3 hours. If x, y, and z together can do ¾ of the work in 3 hours, then x + y + z = (¾)/3 = 3/12 = ¼.
We’re told that y, alone, could have pumped out the rest of the water in 18 hours – again, there was ¾ of a tank left, so y = (¾)/18 = 1/24.
To summarize, we know that x = 1/8, y = 1/24, and x + y + z = ¼; Not so hard to solve for z, right?
1/8 + 1/24 + z = ¼
Multiply everything by 24, and we get:
3 + 1 + 24z = 6
24z = 2
z = 1/12.
That’s z’s rate. If rate and time have a reciprocal relationship, we know that it would take z 12 hours to pump out all the water of one tank alone. The answer is, therefore, B.
Takeaway: The joy of seeing new material from GMAC (Is joy the right word?) is the realization that no matter how many additional layers of complexity the question-writers throw at us, the old verities hold true. So when you see tough questions, slow down. Remind yourself that the strategies you’ve cultivated will unlock even the toughest problems. Then, dive in and discover, yet again, that these questions are never quite as hard as they appear at first glance.