When preparing for the GMAT, you may notice that studying for one subject makes you better in other disciplines as well. For example, practicing your algebra tends to make you better at algebra, arithmetic tends to make you faster at picking numbers and the entire quant section helps you significantly in integrated reasoning. This is due to the fact that many subjects overlap and have common elements. More formally, you can say that the GMAT is an exam with a lot of synergy.
Synergy is defined as “The interaction of two or more agents or forces so that their combined effect is greater than the sum of their individual effects”. The different elements on the exam clearly have some synergy together; however even within specific questions you can notice some elements of synergy that can help simplify the problem.
A good approach when you’re unsure how to attack a problem is to break it down into smaller parts that are easier to digest (like Homer Simpsons’ 6’ sandwich). Instead of trying to figure out everything at once, you break the problem down into more manageable parts and work through them one by one. While this strategy has its upsides, a glaring problem is that you need to recombine the disparate elements back into a cohesive whole. (If you’ve ever taken apart a computer you might know this is sometimes easier said than done). One simple alternative to this piecemeal strategy is to approach questions holistically and consider the entire problem at once.
Let’s examine a problem using both of these strategies:
There are two inlets and one outlet to a cistern. One of the inlets takes 3 hours to fill up the cistern and the other takes twice as much time to fill up the same cistern. If both of the inlets are turned on at 9:00 AM with the cistern completely empty, and at 10:30 AM, the outlet is turned on and it takes 1 more hour to fill the cistern completely, how much time does the outlet working alone take to empty the cistern when the cistern is full?
(A) 2 hours
(B) 2.5 hours
(C) 3 hours
(D) 3.5 hours
(E) 4 hours
Looking at this work-rate problem, we might have to read it two or three times to understand what is going on. There’s a pot of water with two tubes leading in and one leading out (two steps forward, one step back, as it were). The question stem provides a lot of information, so let’s evaluate what we know:
The first inlet takes 3 hours to fill the cistern. The rate is 1/3 of the job per hour.
The second takes twice as long, ergo 6 hours to fill the cistern. The rate is 1/6 of the job per hour.
Ignoring the outlet, what is the rate of both inlets working together? RA + RB = RAB. Mathematically, 1/3 + 1/6 = 6/18 + 3/18 = 9/18 or ½. This means that the two inlets alone complete half the job every hour, and therefore take 2 hours to fill the cistern completely.
Now we can tackle this problem piece-by-piece. If the inlets start at 9:00 AM and the conditions change at 10:30, they had 1.5 hours to fill the cistern. If the rate is ½ per hour and they go for 1.5 hours, then the cistern should be 1.5/2 or ¾ full.
At 10:30, the outlet is turned on and some quantity of water starts to leak out. The cistern is nonetheless full an hour later indicating the inflow of water still outpaces the outflow. The rate of the inlets is known to be ½, but if ¼ of the cistern is filled in 1 hour, then the three streams going simultaneously would take 4 hours to fill the entire cistern. From this, can we determine the rate of just the outlet, as the question is asking?
Algebraically, we can isolate the rate of the outlay:
Rate of Inlet 1 + Rate of Inlet 2 – Rate of Outlet = Rate of all three
1/3 + 1/6 - x = ¼
Putting all the terms on a common denominator (24):
8 / 24 + 4 /24 - x = 6/24
12 / 24 – x = 6 / 24
-x = -6/24
x = 6/24
x = ¼.
The outlay drains ¼ of the cistern per hour, and thus would take 4 hours to drain the entire reservoir. Answer choice E, mathematically proven and clear. However, can we approach this problem holistically and get the same answer faster (oh I hope the answer is yes!)?
If we go back to the two inlets having a combined rate of ½, that means they fill the entire cistern in 2 hours. Adding in the negative effect of the outlay, the rate of the three streams working simultaneously was found to be ¼, meaning the container would be filled in 4 hours. The difference between these two effects is the drain of the outlet. Without the outlet, it takes 2 hours, and with it, it takes twice as long. This means the outlet is draining half the water as it comes in, or, that it has half the rate of the two inlets. Since the two inlets have a rate of ½, the outlet has half of that, or ¼. Still answer choice E, but using the holistic concept instead of algebraic isolation.
Logically, this makes perfect sense and is absolutely correct. There is nothing wrong with using algebra on this question, but a holistic approach will lead to the same exact answer much faster if you understand what is happening conceptually. Breaking down a problem into more manageable pieces is a good strategy that has its place, but taking a holistic approach often helps clarify confusing questions. Just like studying for algebra, geometry and probability makes you better at math in general, using all the elements of a problem often gets you to exploit the inherent synergy of the test.
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.