Last visit was: 17 May 2024, 23:01 It is currently 17 May 2024, 23:01
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

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

SORT BY:
Tags:
Show Tags
Hide Tags
Manager
Joined: 09 Feb 2013
Posts: 104
Own Kudos [?]: 4070 [66]
Given Kudos: 17
CEO
Joined: 24 Jul 2011
Status: World Rank #4 MBA Admissions Consultant
Posts: 3187
Own Kudos [?]: 1591 [15]
Given Kudos: 33
GMAT 1: 780 Q51 V48
GRE 1: Q170 V170
Retired Moderator
Joined: 16 Jun 2012
Posts: 869
Own Kudos [?]: 8574 [12]
Given Kudos: 123
Location: United States
General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 93334
Own Kudos [?]: 624571 [2]
Given Kudos: 81898
Re: There are three different hoses used to fill a pool: hose [#permalink]
1
Kudos
1
Bookmarks
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
Intern
Joined: 29 Jan 2013
Posts: 1
Own Kudos [?]: 1 [1]
Given Kudos: 4
GMAT 1: 490 Q37 V20
GPA: 3.5
Re: There are three different hoses used to fill a pool: hose [#permalink]
1
Bookmarks
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:

$$\frac{c}{3} < d = \frac{abc}{ab+ac+cb}$$
$$c^2b + abc + ac^2 < 3 abc$$
$$c (b+a) < 2ab$$ ???
Retired Moderator
Joined: 16 Jun 2012
Posts: 869
Own Kudos [?]: 8574 [2]
Given Kudos: 123
Location: United States
Re: There are three different hoses used to fill a pool: hose [#permalink]
2
Kudos
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.
Intern
Joined: 05 Jun 2011
Posts: 11
Own Kudos [?]: 23 [1]
Given Kudos: 36
Schools:Kellog, Stern, Stanford, Booth,HBS, Wharton
Re: There are three different hoses used to fill a pool: hose [#permalink]
1
Kudos
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.
VP
Joined: 06 Sep 2013
Posts: 1343
Own Kudos [?]: 2400 [1]
Given Kudos: 355
Concentration: Finance
Re: There are three different hoses used to fill a pool: hose [#permalink]
1
Bookmarks
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
Cheers
J
Manager
Joined: 10 May 2014
Posts: 116
Own Kudos [?]: 340 [0]
Given Kudos: 28
Re: There are three different hoses used to fill a pool: hose [#permalink]
IvanW wrote:
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.

Hi there,
I´m sorry but I dont quite understand the III statement.

- In the question stem we are given that: d < c < b < a
- In III statement we are given that c/3 < d < a/3. Therefore, c < 3d < a

How is it that when you plugged in d = 6, c = 8, and a = 10, the principle held true?

Can you please show me where I am worng?
Board of Directors
Joined: 17 Jul 2014
Posts: 2160
Own Kudos [?]: 1180 [0]
Given Kudos: 236
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE:General Management (Transportation)
Re: There are three different hoses used to fill a pool: hose [#permalink]
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

all together work faster than each individually, thus I is always true, and we can eliminate B and D.
II - same thing as said before, but d can never be greater than individual rate. thus, II is not correct, and we can eliminate E.
III - I used some testing:
a=4, b=2, c=1.
1/4+1/2+1/1 = 1+2+4/4 = 7/4 or d=4/7
c/3 = 1/3
d=4/7
a/3 = 4/3

c/3 < d < a/3
true.
thus, we can eliminate A and select C.
Board of Directors
Joined: 17 Jul 2014
Posts: 2160
Own Kudos [?]: 1180 [0]
Given Kudos: 236
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE:General Management (Transportation)
Re: There are three different hoses used to fill a pool: hose [#permalink]
minwoswoh wrote:
IvanW wrote:
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.

Hi there,
I´m sorry but I dont quite understand the III statement.

- In the question stem we are given that: d < c < b < a
- In III statement we are given that c/3 < d < a/3. Therefore, c < 3d < a

How is it that when you plugged in d = 6, c = 8, and a = 10, the principle held true?

Can you please show me where I am worng?

I don't think you quite understand the concept...you can plug in a, b, and C. D is deducted from a,b,and C. you cannot just plug in value for D.
Manager
Joined: 14 Oct 2012
Posts: 117
Own Kudos [?]: 259 [0]
Given Kudos: 1023
Re: There are three different hoses used to fill a pool: hose [#permalink]
Hello friends, my 2 cents
a>b>c>d => 3>2>1>d
abc/(ab+bc+ca) = 6/(6+2+3) = 6/11 ~ 0.6 ~> 3>2>1>0.6
(iii) c/3 = 1/3 = 0.33 & a/3 = 3/3 = 1 => 0.33 < 0.6 < 1 => c/3 < d < a/3
Target Test Prep Representative
Joined: 04 Mar 2011
Affiliations: Target Test Prep
Posts: 3042
Own Kudos [?]: 6353 [4]
Given Kudos: 1646
Re: There are three different hoses used to fill a pool: hose [#permalink]
3
Kudos
1
Bookmarks
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

Roman numeral I is true. When all three hoses work together, the number of days it takes must be less than the number of days it takes for any individual hose to complete the job by itself. Using the same argument, we see that Roman numeral II can’t be true.

If each hose works as fast as hose z, it will take exactly c/3 days. However, since they do not, and since hose z is the fastest, they must take more than c/3 days. That is, d > c/3. Similarly, if each hose works as fast as hose x, it will take exactly a/3 days. However, since they do not, and since hose x is the slowest, they must take less than a/3 days. That is, d < a/3. We see that c/3 < d < a/3, which means Roman numeral III is true also.

Manager
Joined: 05 Dec 2016
Posts: 91
Own Kudos [?]: 113 [1]
Given Kudos: 184
There are three different hoses used to fill a pool: hose [#permalink]
1
Bookmarks
OFFICIAL VERITAS SOLUTION
Solution: C

Explanation: The best way to solve this question is with a conceptual understanding of work rate relationships (algebra and number picking are both cumbersome). If a, b, and c represent the times of the three hoses, then their time together, d, must be less than the smallest of the three times, c. For instance, if Bill takes $$3$$ hours to mow a lawn, Steve takes $$4$$ hours to mow that lawn, and John takes $$5$$ hours to mow that lawn, together they must take less than $$3$$ hours to mow a lawn! Therefore it must be true that $$d<c$$
so you can eliminate (B) and (D).

The second statement $$d>b$$
must be false for the same logic just given previously (d must be less than each a, b, and c) so you can also eliminate (D).

Lastly, you must decide if statement III must be true. If all three times were the same (say they were all the longest time a) then it would take them $$\frac{a}{3}$$
days to fill the pool. But b and c are faster hoses than a (their times are less) so the time together (d) for the three MUST be less than $$\frac{a}{3}$$
. Likewise, imagine if they are all three the same as the fastest hose with a time of c, then their time would be $$\frac{c}{3}$$
. The other two hoses are slower so the time together (d) must be greater than $$\frac{c}{3}$$
so it must be true that$$\frac{c}{3}<d<\frac{a}{3}$$

. You could prove this with some tedious number picking or some VERY tedious algebra, but the best way is to understand the inverse relationships between times and rates and apply it abstractly.

Manager
Joined: 19 Apr 2017
Posts: 84
Own Kudos [?]: 133 [0]
Given Kudos: 40
Concentration: General Management, Sustainability
Schools: ESSEC '22
GPA: 3.9
WE:Operations (Hospitality and Tourism)
Re: There are three different hoses used to fill a pool: hose [#permalink]
I feel number plugging could be an easier approach for this solution.
If we assign numbers then -> a>b>c could be 8>4>2 (these numbers so the calculations are easier)

To calculate d -> 1/d= 1/8+1/4+1/2 = 7/8 so then d = 8/7

To the question: Which of the following must be true?

I. d<c is 8/7<2 or 8<14 which is true

II. d>b 8/7>4 or 8>28 which is not true

III. c/3<d<a/3 2/3<8/7<8/3 or can be written as 2<24/7<8/3 or can be written as 14<24<56 which is true

so I and III must be true
Intern
Joined: 12 Aug 2019
Posts: 26
Own Kudos [?]: 15 [0]
Given Kudos: 72
Location: India
WE:Engineering (Energy and Utilities)
Re: There are three different hoses used to fill a pool: hose [#permalink]
As given a>b>c
so let a = 10, b = 4, c =2

So the total work done d together : (a*b*c)/(a+b+c) = 80/16 = 5
here, d > b

But b supports the answer in both the cases.
But let us take a = 4, b = 3, c = 2
So total work done: 24/9 = 8/3
now d<b
Manager
Joined: 11 Apr 2020
Posts: 126
Own Kudos [?]: 235 [0]
Given Kudos: 630
GMAT 1: 660 Q49 V31
There are three different hoses used to fill a pool: hose [#permalink]
I didnt want to get into solving algebraic expressions involving work done in 1 day and so on

So I plugged in values (15,10,5), then calculated LCM, did the calculations and checked for relations mentioned and ultimately got this one right praying that there wont be weird values that would destroy my assumption of values/plugging in of values

I want to know, how to make sure (for this question) that the values that I have plugged in are correct

Or

Is plugging values for this one a wrong approach?

Help!
Director
Joined: 09 Jan 2020
Posts: 961
Own Kudos [?]: 226 [0]
Given Kudos: 434
Location: United States
Re: There are three different hoses used to fill a pool: hose [#permalink]
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

If d is the combined rate of 3 different hoses, then d is certainly less than a, b, or c individually. Therefore Statement 1 is true.

If d is less than a, b, or c individually, then Statement 2 can not be true.

For statement 3:

We know a > b > c

Therefore $$\frac{a }{ 3} > \frac{c }{ 3}$$

Since we know d = a + b + c, we can conclude $$\frac{c}{3} < d < \frac{a}{3}$$

Non-Human User
Joined: 09 Sep 2013
Posts: 33057
Own Kudos [?]: 828 [0]
Given Kudos: 0
Re: There are three different hoses used to fill a pool: hose [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: There are three different hoses used to fill a pool: hose [#permalink]
Moderators:
Math Expert
93334 posts
Senior Moderator - Masters Forum
3137 posts