Dealing with rate/work questions

CAT Exam
To review, the key for dealing with rate/work type of question is to apply the following rules:
  1. Rate * Time = Work
  2. Rates are additive in work questions.
  3. 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: 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? A) 6 B) 12 C) 15 D) 18 Okay. Recall 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:  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.

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