Breaking Down GMATPrep Work and Inequalities Problems
Recently at ManhattanGMAT, weve been discussing what kinds of characteristics can turn a medium problem into a hard one, and a hard problem into an almost impossible one. There are many ways to do this, and were going to examine a couple of them in this article.
In the past, weve discussed Rate and Work problems, inequalities, roman numeral problems, and various ways to tackle theory problems (including testing numbers and turning algebra into arithmetic). This week, weve got a seriously difficult problem from GMATPrep that touches on all of these areas, though I would say that its primarily a Work problem at heart.
Heres the GMATPrep problem. Set your timer for 2 minutes. and GO!
In a certain bathtub, both the cold-water and the hot-water fixtures leak. The cold-water leak alone would fill an empty bucket in c hours, and the hot-water leak alone would fill the same bucket in h hours, where c < h. If both fixtures began to leak at the same time into the empty bucket at their respective constant rates and consequently it took t hours to fill the bucket, which of the following must be true?I. 0 < t < h
II. c < t < h
III. [pmath]c/2<t<h/2[/pmath]
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III
Done? You can take up to an extra 30 seconds if you want, but ONLY if you know exactly what youre doing. If you want an extra 30 seconds because youre thinking that maybe you could get it if you just had a little more time stop now and pick and answer.
There are two kinds of leaks: a Cold leak and a Hot leak. What do we know about each? Draw a picture if that helps you (it certainly helps me).
The Cold leak fills 1 bucket in c hours. Given the formula RT=W, or rate time = work, we can determine that R = W/T = 1/c. For future reference, if you know that something can complete 1 job in x hours, then the rate is always just 1/x. (Note: in this problem, the job = filling the bucket.)
Okay, so we have a rate for the Cold leak. And we can calculate the same for the Hot leak! The rate for the Hot leak is simply 1/h.
They also tell us that c < h. Those two variables represent how much time it takes for the bucket to fill, so this is saying that length of time c is lower than length of time h. So which leak is leaking faster which leak has the higher rate? The Cold bucket fills up sooner, so the Cold leak is leaking at a higher rate. The time is lower and the rate is higher.
In the final sentence, were given a scenario where both leaks begin leaking at the same time, and both leaks contribute to filling a bucket. The two leaks are said to be working together to finish the job. That job takes t hours, so the rate of both together is 1/t.
At this point, most people would likely begin to do one of two things:
(1) plug in some real numbers for the various variables given in the problem and try to turn the algebra into arithmetic, or
(2) start writing equations, such as rate of C alone + rate of H alone = combined rate of the two working together or [pmath] 1/c+1/h=1/t[/pmath]
On many problems, those approaches might be good ones but they dont turn out to work that well on this particular problem. How could we know that before doing the problem? We havent actually finished examining the problem yet we still have the question itself and the roman numerals. Lets see how that additional information changes the game.
First, the question asks us to figure out which of three different inequalities MUST be true. We can pick real numbers on MUST be true theory problems, but it is often the case that we need to try multiple numbers or combinations of numbers in order to prove whether something is always true versus just sometimes true, and efficiency also requires us to have a pretty good idea of the different kinds of numbers or combinations that might make a difference. Take a look at those roman numerals. Do you want to test multiple cases? Me neither.
Next, we could write a working together equation from the information given above, but what are we trying to find? Were not trying to solve for one particular value. Were not even trying to solve for one variable in terms of the other variables. Weve got inequalities in the roman numerals. The question stem gave me one inequality, which is nice but my other piece is an equation. How do I combine and try to create the roman numeral inequalities? If I cant figure out how to create them from my starting equation and inequality, does that mean they dont have to be true, or does it just mean that I havent figured out the right manipulations?
So where do we go from here? Lets try to get a better sense of what theyre trying to test. What does the first roman numeral inequality mean in normal English? (Note: in general, start with whatever roman numeral you think is easiest; in this case, I and II both seem better than III to me!)
I. 0 < t < h
So this says: t, the time it takes for the two when they work together, is shorter than h, the time it takes when the hot water leak is leaking all by itself.
Is that ever true? If so, is it true sometimes or all of the time? Go take a look at your picture again, if you drew one, and visualize whats going on. The Cold and Hot leaks each take some amount of time to fill the bucket when leaking separately. If both leaks start up at the same time, then its going to take less time to fill the bucket than it would take if only one was leaking. So, yes, its always true that the time it takes to fill the bucket when theyre both leaking is shorter than the time it takes to fill the bucket when only the Hot leak is leaking. (And, of course, it makes sense that the times are positive, or greater than zero.)
Roman numeral I is always true. Eliminate wrong answers next. In this case, we can eliminate B and C.
Okay, next! I really dont want to deal with III unless I have to, so Im moving to II next.
II. c < t < h
Lets try that same tactic: what does this mean in normal English? Also, break it down into parts. First, weve got c < t. The time it takes for the Cold leak to fill the bucket is shorter than the time it takes for Cold + Hot leaks working together to fill the bucket.
Hmm. That doesnt make a lot of sense. So, the Cold leak is leaking and then I add the Hot leak as well and then it takes longer to fill the bucket? No. Roman numeral II is false. Eliminate answer D.
Now we get to the really nasty statement. At this point, if youre already pushing 2+ minutes, consider guessing between A and E and moving on. If your time is okay and you feel comfortable with everything else youve had to do so far, then try III.
III. [pmath]c/2<t<h/2[/pmath]
You know what to do: what does this mean in normal English?
Start with the first part: [pmath]c/2<t[/pmath]. The time it takes the Cold leak by itself to fill HALF of a bucket is less than the time it takes the two leaks together to fill one entire bucket. Thats weird. They just changed the scenario now Im comparing the time it takes for c to fill half of a bucket by itself, not a whole bucket. Ill have to remember, for future, that they might do something like this.
This next part is a leap that you probably wouldnt figure out spontaneously in 5 seconds during a real question. It requires some thought and playing around with concepts so this would be something you figure out during practice and then you remember to apply it when you see a similar situation on a future question.
When the two leaks, Cold and Hot, are working together to fill the one bucket, which leak is going to be responsible for filling more of the bucket, Cold or Hot?
Cold, because thats the faster leak. So visualize that one full bucket sitting there, filled by both leaks together. More than half of the water came from the Cold leak and less than half of the water came from the Hot leak, right?
And how long did it take for the Cold leak to fill more than half of that bucket? Time t, because thats how long both leaks worked together. So would it take more or less time for the Cold leak, working by itself, to fill only half of that bucket? It would take less time always. The first half of the inequality is okay.
What about the second half: [pmath]t<h/2[/pmath]?
Use the same reasoning. First, translate: the two leaks together take less time to fill an entire bucket than the Hot leak takes by itself to fill half of a bucket. (Draw that out visually if you think it will help!)
The Hot leak has filled up less than half of the bucket during time t (because the Cold leak filled up more than half, right?). So it would take the Hot leak, working by itself, some longer time than that to fill up half of the bucket. Hey, this half must be true all the time as well! Statement III must be true. Eliminate answer A.
The correct answer is E: Statements I and III must be true.
Key Takeaways for Problem Solving Work + Inequalities Theory Problems:
(1) Very difficult problems will combine content from multiple math disciplines. They may also seem to point to certain solution techniques (such as, in this case, picking numbers or solving equations). Look more closely at the details: know how to recognize when certain "standard" approaches really wouldn't be a good fit.
(2) On must be true questions, inequalities are often used as code for certain principles or characteristics that they want us to be able to derive (for example: xy < 0 means that x and y have opposite signs). These principles are often easier to derive if we do so using English and logic rather than math (though note that easier does not mean easy; this is still a very challenging problem).
(3) Review and analysis will be necessary in order to use this what does it mean in English technique efficiently and effectively. The first two roman numerals here were doable in a timely fashion if you already felt comfortable with the technique, but the third one is a lot harder. If you had been exposed to something like it before and really studied it, then it would (hopefully!) spring to mind again when you saw something similar on this new problem.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.