You should be careful using this approach on "MUST be true" problems. If you don't know how to do it any other way, at least try a few different values to increase your confidence in the result. The danger is that if the statement you are testing is only sometimes true, you might pick values that happen to make it work, which might make you think it is always true, and you won't see the values that make it untrue. For example, "Is 2^n+1 always prime for every non-negative integer n?" Here, you might plug in 0, 1, and 2 and get 2,3, and 5 and conclude that it is, in fact, always prime. But if you plug in n=3, you get 9, which is not prime. So the statement is not always true. So you generally want some other evidence to convince yourself that something is ALWAYS true.
Another way to analyze this problem is as follows:
I. 0<t<h The time will always be less than h because h is how long it takes with the hot water fixture alone, so with the help of c, the bucket will fill up faster. And, obviously, the time cannot be zero or negative. ALWAYS true.
II. c<t<h. This is never true because it takes c hours with just the cold water fixture, so with the help of the hot water fixture, the bucket must be filled up in less time than c hours. NEVER true.
III. c/2<t<h/2. c/2 is how long it would take if both fixtures were leaking as fast as the cold water fixture. But the hot water fixture is leaking more slowly than the cold, so it will take more time than if we had 2 fixtures both leaking at the cold water rate. Hence, t>c/2. On the other hand, h/2 is how long it would take if we had two fixtures both leaking at the rate of the hot water fixture. But we actually have replaced one of those with the faster leaking cold water fixture, so it must take less time than h/2 hours. So t<h/2. Combining, it must be true that c/2<t<h/2. ALWAYS true.
Whoever wrote this problem must have been a nice guy, because he didn't create any trap answers: statements that are sometimes true, but are not always true. For example, let's say we had to evaluate whether c/3<t<h/3 is always true. If we use c=2 and h=10, then t=5/3, and 2/3<5/3<10/3, so it appears to be true. But if we choose c=2 and h=3, then t=6/5, and 2/3<6/5<1 is NOT true, so the statement is not always true, but we didn't see that until the second round of plugging in values.
So, again, be careful with this strategy on MUST be true questions. If you do use the strategy, try to get some supporting evidence for why the statements are ALWAYS true. If you can't see the logic behind it, at least try plugging in a range of values, time permitting. In a case like this, it might mean plugging in values of c and h that are both small or both large or close together or far apart. In other types of problems it might mean making sure you try out values that are negative or fractions, or zero.
