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. c/2 '<' t '<' h/2
A) I only B) II only C) III only D) I and II E) I and III



when the two taps are working together, then the time to fill the bucket, t, must always be less than either tap working alone, so I must be true. By exactly the same reasoning, II cannot be true, so we can rule out answer choices B,C and D.

This problem is one where if you go down the "plug in numbers" approach you really need to pick your numbers carefully - don't forget that you're told c'<'h in the problem statement. E.g. c=2, h=3 then t=6/5 and it is true that 2/2'<'6/5'<'3/2. Another example c=5, h=15 then t= 15/4 and it'strue that 5/2'<'15/4'<'15/2. So this might give you enough confidence to conclude that III is always true and that E is the answer.
If you're quick and confident with the algebra you can answer this question definitively though. When the two taps are filling bucket together:
1/t = 1/c + 1/h 1/t = (c+h)/ch t=ch/(c+h)
Let's consider the right side of inequality III (t '<' h/2). Substiture the expression for t above:
ch/(c+h)'<'h/2
c/(c+h)'<'1/2
is this always true when c'<'h? It sure is, since c+h must be greater than 2c
Ans - E