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Hi, there. I'm happy to help with this.

Prompt:
In a certain bathtub, both the cold-water and the hot-water fixtures leak. The cold-water leak alone with fill an empty bucket in c hours, and the hot water 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 the 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
111. c/2 < t < h/2
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

So, first of all, without doing any math, we know that the hot water is a slower leak, because it would take longer to fill the same bucket that the cold water would fill in a shorter time. We also know that hot & cold together will fill the bucket faster than either hot alone or cold alone, so clearly 0 < t < h and 0 < t < c. The first is statement I and the second is not an option. We know statement I has to be true, so right away, (B) and (C) are out. The second statement implies that II is false --- t must be smaller than c!! --- so (D) is out also. If you were stuck on everything else about the problem, at least you could guess from the remaining three answers, and the odds would be in your favor (the odds are always in your favor when you eliminate one or more answer choices).

Now, how to solve this.

First, convert the information to rates:
Cold water fills one bucket in c hours --- rate of cold = Rc = 1/c
Hot water fills one bucket in h hours --- rate of hot = Rh = 1/h

We know h > c, so we know 1/h < 1/c ---- i.e. Rh < Rc --- this is what we said above: cold must be leaking faster.

Combined, hot & cold fill one bucket in t hours ---- total combined rate = Rt = 1/t

BIG IDEA -- we never add or subtract times that it take things to accumulate --- we add rates.

We get the equation 1/t = 1/h + 1/c. We would like t (or one over t) in the middle of a three part inequality, to compare to statement III. Consider this.

Begin the statement 1/h < 1/c
Add 1/h to both sides ----> 2/h < 1/h + 1/c (the right side of this is 1/t)
Instead, add 1/c to both sides ----> 1/h + 1/c < 2/c (the left side of this is 1/t)
Combine those two inequalities
2/h < t < 2/c
Take the reciprocal of every term, which reverses the order of the inequalities:
h/2 > t > c/2
That's statement III, which must be true.

We already determined that statement I must be true, so the answer is E.

That's a very trick problem. Does that make sense?

Here's a similar problem ---
http://gmat.magoosh.com/questions/47
When you submit your answer to this question, the next page will have a full video explanation.

Let me know if you have any further questions.

Mike

_________________
Magoosh GMAT Instructor
http://gmat.magoosh.com/