How to work on rate problems

Rate questions, so far as I can remember, have been a staple of almost every standardized test  ever. And, rest assured, dear reader, you will see them on the CAT. When dealing with a complex work question there are typically only two things we need to keep in mind, aside from our standard “rate * time = work” equation. First, we know that rates are  additive. If I can do 1 job in 4 hours, my rate is 1/4. If you can do 1 job in 3 hours, your rate is 1/3. Therefore, our combined rate is 1/4 + 1/3, or 7/12. So we can do 7 jobs in 12 hours. The second thing we need to bear in mind is that rate and time have a reciprocal relationship. If our rate is 7/12, then the time it would take us to complete a job is 12/7 hours. Not so complex. What’s interesting is that these simple ideas can unlock seemingly complex questions. Take this official question, for example: Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank. A) 1/3 B) 1/2 C) 2/3 D) 5/6 E) 1 So let’s start by assigning some variables. We’ll call the rate for p ump A, R_a. Similarly, we’ll designate the rate for pump B as R_b,and the rate for pump C as R_c. If the time for A and B together to fill the tank is 6/5 hours, then we know that their combined rate is 5/6, because again, time and rate have a reciprocal relationship. So this first piece of information yields the following equation: R_a + R_b = 5/6. If A and C can fill the tank in 3/2 hours, then, employing identical logic, their combined rate will be 2/3, and we’ll get: R_a + R_c = 2/3. Last, if B and C can fill tank in 2 hours, then their combined rate will be ½, and we’ll have: R_b+ R_c = 1/2. Ultimately, what we want here is the time it would take all three pumps working together to fill the tank. If we can find the combined rate, or R_a + R_b + R_c, then all we need to do is take the reciprocal of that number, and we’ll have our time to full the pump. So now, looking at the above equations, how can we get R_a + R_b + R_c on one side of an equation? First, let’s line our equations up vertically:  R_a + R_b = 5/6. R_a + R_c = 2/3. R_b+ R_c = 1/2.  Now, if we sum those equations, we’ll get the following: 2R_a + 2R_b + 2R_c = 5/6 + 2/3 + 1/2. This simplifies to: 2R_a + 2R_b + 2R_c = 5/6 + 4/6 + 3/6 = 12/6 or 2R_a + 2R_b + 2R_c  = 2. Dividing both sides by 2, we’ll get: R_a + R_b + R_c  = 1. This tells us that the pumps, all working together can do one tank in one hour. Well, if the rate is 1, and the time is the reciprocal of the rate, it’s pretty obvious that the time to complete the task is also 1. The answer, therefore, is E. Takeaway: the most persistent myth we have about our academic limitations is that we’re simply not good at a certain subset of problems when, in truth, we just never properly learned how to do this type of question. Like every other topic on the CAT, rate/work questions can be mastered rapidly with a sound framework and a little practice. So file away the notion that rates can be added in work questions and that time and rate have a reciprocal relationship. Then do a few practice questions, move on to the next topic, and know that you’re one step closer to mastering the skills that will lead you to your desired CAT score.