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 - Answer

I picked numbers for c (3 hours) and h (6 hours); then, I found t (2 hours) and got my answer that way. Is there a better, more systematic way for solving this type of work problem?

I think the answer should be A.

If you pick different values of c & h, statement 1 always is true while statement III can be true depending upon the values picked.


Statement II cannot be true in any scenario based upon the parameters set by the question.

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.

As for the left side of the inequality (c/2 <t) you can see that this is true intuitively by noting that since h takes longer to fill the bucket by itself than c, it can't cut c's time in half when they are working together. You can also see it algebraically:

c/2 < ch/(c+h)

1/2 < h/(c+h)

if c<h, then this must always be true sincec+h<2h.

so I and III must always be true and the answer is E.

Jeff

Jeff, well done with the algebra! For those of you less comfortable with the algebraic solution, plugging in numbers is a decent secondary approach. To increase your chances of viewing a diversity of answers, choose numbers for c and h that differ in size and proximity to each other (e.g. small c, small h; large c, large h; c and h close together; c and h further apart).

It looks like it's been a year since anyone has discussed this problem. Can anyone provide a more solid dumbed down explanation as to why III. makes sense?

absolutely.
i'm pretty dumb at heart, so i loves me my dumbed down explanations.

here's the deal:
background fact: if you have two things doing the same work at the same rate, then they can do it in half the time. <-- this fact is all you need to understand the following explanation.

the faster leak takes C hours and the slower leak takes H hours.

* if both buckets were as fast as the fast one, then they would take C/2 hours.
--- but they aren't, so they take longer than that.
* if both buckets were as slow as the slow one, then they would take H/2 hours.
--- but they aren't, so they're faster than that.

done.