Time and work problems

The rate and work questions  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! Try this problem out: * 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   Okay.Not just two but three separate pumps, all presumably operating at different rates. Some pretty ugly numbers. This is not going to be fun. 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. 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 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.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 or skip it. 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. J (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!