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There are three different hoses used to fill a pool: hose

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There are three different hoses used to fill a pool: hose [#permalink] New post 11 Jun 2013, 03:31
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There are three different hoses used to fill a pool: hose x, hose y, and hose z. Hose x can fill the pool in a days, hose y in b days, and hose z in c days, where a > b > c. When all three hoses are used together to fill a pool, it takes d days to fill the pool. Which of the following must be true?
I. d<c
II. d>b
III. c/3<d<a/3

A) I only
B) III only
C) I and III only
D) II only
E) I, II and III
[Reveal] Spoiler: OA

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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 11 Jun 2013, 03:59
Expert's post
emmak wrote:
There are three different hoses used to fill a pool: hose x, hose y, and hose z. Hose x can fill the pool in a days, hose y in b days, and hose z in c days, where a > b > c. When all three hoses are used together to fill a pool, it takes d days to fill the pool. Which of the following must be true?
I. d<c
II. d>b
III. c/3<d<a/3

A) I only
B) III only
C) I and III only
D) II only
E) I, II and III


"Stolen" question from GMAT Prep:
Quote:
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


Discussed here: in-a-certain-bathtub-both-the-cold-water-and-the-hot-water-127878.html
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 11 Jun 2013, 22:30
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Name T = full pool
X fills a pool in a days ==> 1 day X fills: T/a
Y fills a pool in b days ==> 1 day Y fills: T/b
Z fills a pool in c days ==> 1 day Z fills: T/c

1 day (X+Y+Z) together fill: T(1/a + 1/b + 1/c)
d days (X+Y+Z) together fill: T

==> d = Tx1 / T(1/a+1/b+1/c) = abc/(ab+bc+ca)
==> d = abc/(ab+bc+ca)

Statement 1: d < c ==> Correct because three hoses together fill faster than one hose does

Statement 2: d > b ==> Wrong because d may be less than or greater than b. Please note that the question is MUST BE TRUE.

Statement 3: c/3 < d < a/3 ==> Correct

* Because (ab+bc+ca) < 3ab. [Please note that a > b > c]
==> d = abc/(ab+bc+ca) > abc/3ab ==> d > c/3

* Because (ab+bc+ca) > 3bc [ab > bc; bc = bc; ac > bc ==> ab+bc+ca > 3bc]
==> d = abc/(ab+bc+ca) < abc/3bc ==> d < a/3

Hence, C is correct.
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 20 Jun 2013, 05:32
pqhai wrote:
Name T = full pool
X fills a pool in a days ==> 1 day X fills: T/a
Y fills a pool in b days ==> 1 day Y fills: T/b
Z fills a pool in c days ==> 1 day Z fills: T/c

1 day (X+Y+Z) together fill: T(1/a + 1/b + 1/c)
d days (X+Y+Z) together fill: T

==> d = Tx1 / T(1/a+1/b+1/c) = abc/(ab+bc+ca)
==> d = abc/(ab+bc+ca)

Statement 1: d < c ==> Correct because three hoses together fill faster than one hose does

Statement 2: d > b ==> Wrong because d may be less than or greater than b. Please note that the question is MUST BE TRUE.

Statement 3: c/3 < d < a/3 ==> Correct

* Because (ab+bc+ca) < 3ab. [Please note that a > b > c]
==> d = abc/(ab+bc+ca) > abc/3ab ==> d > c/3

* Because (ab+bc+ca) > 3bc [ab > bc; bc = bc; ac > bc ==> ab+bc+ca > 3bc]
==> d = abc/(ab+bc+ca) < abc/3bc ==> d < a/3

Hence, C is correct.


Thanks for the explanation, however, I don't understand how you get this inequality: (ab+bc+ca) < 3ab

I tried to solve it but was unsuccesful :(

Here is where I got stuck:

\frac{c}{3} < d = \frac{abc}{ab+ac+cb}
c^2b + abc + ac^2 < 3 abc
c (b+a) < 2ab ???
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 20 Jun 2013, 11:00
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szDave wrote:
pqhai wrote:
Name T = full pool
X fills a pool in a days ==> 1 day X fills: T/a
Y fills a pool in b days ==> 1 day Y fills: T/b
Z fills a pool in c days ==> 1 day Z fills: T/c

1 day (X+Y+Z) together fill: T(1/a + 1/b + 1/c)
d days (X+Y+Z) together fill: T

==> d = Tx1 / T(1/a+1/b+1/c) = abc/(ab+bc+ca)
==> d = abc/(ab+bc+ca)

Statement 1: d < c ==> Correct because three hoses together fill faster than one hose does

Statement 2: d > b ==> Wrong because d may be less than or greater than b. Please note that the question is MUST BE TRUE.

Statement 3: c/3 < d < a/3 ==> Correct

* Because (ab+bc+ca) < 3ab. [Please note that a > b > c]
==> d = abc/(ab+bc+ca) > abc/3ab ==> d > c/3

* Because (ab+bc+ca) > 3bc [ab > bc; bc = bc; ac > bc ==> ab+bc+ca > 3bc]
==> d = abc/(ab+bc+ca) < abc/3bc ==> d < a/3

Hence, C is correct.


Thanks for the explanation, however, I don't understand how you get this inequality: (ab+bc+ca) < 3ab

I tried to solve it but was unsuccesful :(

Here is where I got stuck:

\frac{c}{3} < d = \frac{abc}{ab+ac+cb}
c^2b + abc + ac^2 < 3 abc
c (b+a) < 2ab ???


Hi szDave
Because a > b > c
So:
(1) ab = ab
(2) ab > bc (because a > c ==> a*b > c*b)
(3) ab > ca (because b > c ==> b*a > c*a)

(1) + (2) + (3) = 3ab > ab + bc + ca
This is the key for this question.

Hope it helps.

Regards.
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 26 Sep 2013, 08:32
Quote:
There are three different hoses used to fill a pool: hose x, hose y, and hose z. Hose x can fill the pool in a days, hose y in b days, and hose z in c days, where a > b > c. When all three hoses are used together to fill a pool, it takes d days to fill the pool. Which of the following must be true?
I. d<c
II. d>b
III. c/3<d<a/3

A) I only
B) III only
C) I and III only
D) II only
E) I, II and III


I used a conceptual PLUS VIC ( variables in choices MGMAT) approach.

I. d<c - We KNOW this is True because 3 hoses working together MUST BE faster than one hose by itself! B&D out!!
II. d>b - This is conceptual as well because we can think of many instances where combining 3 hoses/machines etc. would be faster than ANY individual machine, that's kinda the benefit of combining your rates to increase efficiency so....D & E are out!!

Now we have a 50/50 chance between A & C! better than 20% eh?

III. c/3<d<a/3 - With this option I knew I could try it algebraically but it's very easy to get tangled up in "alphabet soup" (for me), so I went with VIC! **Plus the thing about MUST BE TRUE options is that all you have to do is find 1 option that is to the contrary and you are good to go!** I plugged in 10, 8, and 6, but you can plug in any values and you will see that the principal holds.
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 31 Mar 2014, 08:24
Stolen question from GMAT Prep again. Prep companies need to start getting more creative

Express solution

Statement 1 is correct because the three hoses together fill faster than one hose does ALWAYS.

Now, the second statement is NOT always true because 'd' may be less than or greater than 'b', it depends on the values of 'a' and 'c'.

Third condition is ALWAYS true, 3 times the average must always be between the extremes.

Therefore C is correct

Hope this helps
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Re: There are three different hoses used to fill a pool: hose [#permalink] New post 31 Mar 2014, 10:48
This question can be answered without really resorting to any calculations at all.

I. d<c. This has to be true, because the three hoses working together will take less time to fill the pool than hose z working alone. CORRECT.
II. d>b. This has to be false, because the time taken by the three hoses working together cannot be more than the time taken by hose y working alone. INCORRECT.
III. c/3<d<a/3. If all three hoses worked at the rate of the slowest (i.e. hose x which takes a days), then the time taken to fill the pool would be a/3. Since the other two hoses (y and z) are faster than x, the time taken has to be less than a/3. So d<a/3.
If all three hoses worked at the rate of the fastest (i.e. hose z which takes c days), then the time taken to fill the pool would be c/3. As the other two hoses (x and y) are slower than z, the time taken has to be more than c/3. So d>c/3.
CORRECT.

(C) it is.
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Re: There are three different hoses used to fill a pool: hose   [#permalink] 31 Mar 2014, 10:48
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