Hi All,

We're told that In a certain bathtub, both the hot and cold water fixtures leak. The cold water leak alone would fill an empty bucket in C hours, and the hot water leak alone will 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. We're asked which of the following MUST be true. Based on the five answer choices, we know AT LEAST one of the three Roman Numerals is always true - and we can TEST VALUES to define which are always true and which are not always true.

To start, this is an example of a Work Formula question, so we can use the Work Formula:

(A)(B)/(A+B) = time it takes to complete the task together, where A and B are the individual times needed to complete the task. In the prompt, we're told that C < H, so we can TEST C = 3 hours, H = 6 hours... meaning that the TOTAL time to fill the bucket would be (3)(6)/(3+6) = 18/9 = 2 hours... so T = 2. With those three values, we can check the Roman Numerals...

I. 0 < T < H

With our values, T = 2 and H = 6... and 0 < 2 < H, so Roman Numeral 1 appears to be true. Logically, we can also deduce that Roman Numeral 1 will ALWAYS be true, since when BOTH fixtures leak, the amount of time needed to fill the bucket would obviously be SMALLER than if just one of the fixtures was leaking. This means that T < H and T < C will ALWAYS be true and all of those variables will be greater than 0.
Eliminate Answers B and C.

II. C < T < H

With our values, C=3, T = 2 and H = 6... but 3 < 2 < H is NOT true, so Roman Numeral 2 is NOT true
Eliminate Answer D.

III. C/2 < T < H/2

With our values, C=3, T = 2 and H = 6... and 3/2 < 2 < 6/2 IS true, so Roman Numeral 3 appears to be true. Roman Numeral 3 will also ALWAYS be true, but you would have to do a bit more work to prove it. With ANY pair of C and H that fits the given parameters, we'll end up with a T that is less than both. Since C is the faster rate (in this example, 3 hours to fill a bucket is faster than 6 hours to fill a bucket), if we divide C by 2 and H by 2, then those rates DOUBLE (it would then take just 1.5 hours and 3 hours, respectively, to fill the bucket). That clearly gives us one values that is LESS than the current value of T and one that is MORE than the current value of T.
Eliminate Answer A.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

We can easily use logic to get the correct answer, without doing any calculations.

In WORK rate problems, when two machines / faucets / people, etc, work together,
a) they work at the SUM of their respective rates, and
b) the time that it takes them to complete the task together is LESS than the time it took either one individually

Thus, the time t for both leaks working together must be less than both c and h. This eliminates II.

Time must always be positive, so roman numeral I. works.

For III, think of it this way: if two machines both completed a given task in x hrs individually, then when they worked together it would take half the time: x/2. So for our purposes here, imagine that 2 cold faucet leaks would take c/2 hrs... but one cold + one hot would take slightly longer, because the hot faucet leak takes longer. Likewise, 2 hot faucet leaks would take h/2 hrs... but one hot + one cold will take less than that.

Thus, the time t must be between c/2 and h/2. Number III works.

The answer is E.

For more on using logic to estimate the time 2 things spend working together, see: https://www.beatthegmat.com/tank-fill-t ... tml#757795

swerve wrote:In a certain bathtub, both the hot and cold water fixtures leak. The cold water leak alone would fill an empty bucket in c hours, and the hot water leak alone will 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

We can create the equation:

1/c + 1/h = 1/t

If we let c = 2 and h = 3, we have:

1/2 + 1/3 = 3/6 + 2/6 = 5/6.

So we see that t = 1/(5/6) = 6/5 = 1.2.

So we have t < c < h, so statement I is correct and statement II is not.

Let's now analyze statement III.

c/2 = 2/2 = 1

h/2 = 3/2 = 1.5

Thus:

c/2 < t < h/2

Answer: E