This Problem is Work!

by on April 5th, 2013

Raise your hand if you love rate and work questions. They’re awesome, right? They tend to be fairly long, and the set-up is pretty complex, plus we get to build a table before we dive into the equations!

Oh, wait… no… those are all reasons why we can’t stand these problems.

Give yourself approximately 2 minutes to try the below GMATPrep® problem. When you’re done, take a look at it again and ask yourself, “Is there a better way to do this thing?”

* “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”

Have you got an answer? Pick one anyway. Pretend it’s the real test: you can’t keep going till you pick an answer.

(Aside: why am I insisting that you pick an answer? It doesn’t matter – this isn’t the real test! Actually, it really does matter. If you can’t make yourself pick a random answer right now, then you’re going to hang on too long and waste time during the real test. Practice making yourself let go.)

Okay, back to our problem. Yuck. Not just two but three separate pumps, all presumably operating at different rates. Some pretty ugly numbers. This is not going to be fun.

Hmm. It seems like we need to figure out the individual rates of the three pumps. The three pieces of information would let us write three equations, and we’d have three variables, so we could use simultaneous equations rules to find the rates for all three pumps. Then we can add them up to find out how long they take to do the job working together.

Let’s see. First, let’s label each pumps rate according to the pump name. Pump A’s rate is A, pump B’s rate is B, and pump C’s rate is C.

Together, A and B fill the tank in 6/5 hours; that is A’s time and B’s time add up to 6/5 hours. Remember the standard “work” formula: RT = W. If they take 6/5 hours to do 1 job, then their combined rate is R(6/5) = 1, or R = 5/6. As a shortcut, know that if a machine takes a certain length of time to do 1 job, you can just take the reciprocal of that time to find the rate. In this case, 6/5 hours to do the job becomes a rate of 5/6 per hour.

Pump A’s rate and pump B’s rate, then, add up to a combined rate of 5/6, for our first equation:

(EQN1) A + B = 5/6

Do the same thing with the other two pieces of info to get two more equations:

(EQN2) A + C = 2/3

(EQN3) B + C = 1/2

All right! We’re getting somewhere. Now, our next step is to figure out how to solve for A, B, and C.

At this point, people will have used at least 30 seconds and probably closer to 60 – and the really hard math is still to come. Who thinks they’re going to get through the rest of it in another minute? Okay, honestly, I probably would… but I really don’t want to. :) There’s got to be a better way!

What are we solving for again? We want to find the combined rate for all three pumps, or A + B + C. Instead of finding each one individually, can you think of a way to solve for that “combination”: A + B + C?

If you’re not sure, take a minute to think about it before you keep reading.

Okay, try this: add all three equations together at once. We’ll end up with 2A + 2B + 2C = (5/6) + (2/3) + (1/2). Start simplifying.

2A + 2B + 2C = (5/6) + (4/6) + (3/6)

2A + 2B + 2C = 12/6

A + B + C = 6/6 = 1

If all three work together, they’ll get the job done in 1 hour.

The correct answer is E.

What if you can’t get through the math? How could someone make a reasonable guess on this problem. Take a little time to try to figure something out and then come back here and keep reading.

If A and B do the job in 6/5 of an hour, but we knock out pump B and substitute C instead, suddenly it takes 1.5 hours, so C must be slower than B. Likewise, A and C take 1.5 hours, but replace A with B and the time jumps to 2 hours, so B must be slower than A. The rates then are in this order: C < B < A.

Further, pump C must be significantly slower than A. It takes A and B about an hour to do the job, while B and C take 2 hours. Pump B is the same, but A swaps out for C and the time doubles!

The two faster pumps, A and B, need a little over an hour to finish the job. If the slowest, C, is added, that’ll reduce the overall time. None of the answers is over 6/5 – too bad. But keep thinking. If it takes the two faster ones a little over an hour to do the job, then adding the slowest pump isn’t going to speed things up all that much. For instance, the slowest pump won’t suddenly drop the entire job to only 1/3 of an hour! If that happened, then C would have to be the fastest pump.

Further, C has to be significantly slower than A, remember? So even 1/2 is probably too high. The answer is likely either C, D, or E. Pick one and move on.

Key Takeaways for Annoying Work Problems

(1) Sure, you can always set up those charts, write out all of the equations, and solve for each variable individually. But isn’t there an easier way?!? Yes, most of the time there is. Study these enough in advance that you learn a bunch of shortcuts.

(2) For instance (and this applies to any algebra problem, not just a Work problem), when they’re asking us to solve for a combination of variables (in this case, A + B + C), consider whether you can solve for the whole combo at once rather than finding each individual value. Maybe you can’t – but at least think about it. :)

(3) Don’t forget to study how to guess! Do you want to spend 4 minutes doing all of the annoying math on this one (with a high likelihood of making a mistake, because there’s so much math to do)? Or do you want to admit that you haven’t found the easier way to do it and take 2 minutes to guess and move on? In this case, you could probably even save a little bit of time and still give yourself a 1/3 chance of getting it right!

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

7 comments

  • For some students going through the algebraic way is much easier as there is not much logical thinking to be done and one can almost mechanically crack the problem in 90 seconds. However for others who are better at logic than at algebra your second approach will be better.

  • I completely agree with AbhiJ. I almost every time do Work-related problem with algebra. Some might find the other way more fascinating. 

  • There are multiple ways to do any quant problem and no one "right" way - just depends on how your brain works (as well as your own strengths and weaknesses)!

    I'll just point one thing out - many of the toughest math problems have what we (in the industry) call "elegant" solutions. These are solutions that are logic- or theory-based and can be done very quickly and easily - IF, of course, you thoroughly  understand the logic or theory. If you want an absolute top score on the test, challenge yourself to learn these approaches.

    But that level is overkill for many, if not most, test-takers! :)

    • Hi Stacey,

      Thanks for pointing out the reason behind going for logical solution. Its a matter of developing logical intuition(doing things in your head). I was amazed at one guy who was able to solve quant problems in like 20-30 seconds, the problems which take 2-3 mins by the normal approach.

    • I totally agree with Stacey. In order to break 700+ barrrier, we do not have more than 2 minutes to solve each problem. I like the way she called those questions with "elegant" solutions. It is indeed true. Learning multiple approaches to solve the GMAT problems are good but sometimes at the end of day the logic or theory way will get you ahead of the game.

      Charu and AbhiJ are also correct. We do feel comfortable using algebra approach to solve problems. However, we also should learn the new approaches instead of sitting comfortably on "traditional" ways. When we all go thru MBA program, we will be learning all new sorts of ways. In my opinion, why not start now as we study for the GMAT.

    • Hi,

      I generally find it easier to go by %tages to solve most of the work rate problems. Like in this case. I solve this question almost orally without flexing the brain much.

      Here's how I did it.

      A+B= 5/6 x 100 = 83%odd (this in turn gives me the efficiency of these two machines working together i.e. they complete 83% of the work in 1 hour together). 
      A+C= 2/3 x 100= 66% odd
      B+C =1/2x 100 =50%. 
      Adding all the three 2(A+B+C)= 200 (approx)
      A+B+C=100% (i.e they complete the whole work in 1 hour.) Hence the answer is 1hr E

    • Kelly, I love that attitude. :)

      Vaibhav, yes - you basically used the same solution process I describe in the article. I used the given fractions and you converted to percentages, but either is fine. That's why we put Fractions, Decimals, and Percents all together in the same book!

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