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Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 24% (03:18) correct 76% (02:56) wrong based on 38 sessions

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Question of the Week #43

While working individually, each of A, B, C and D can produce 60 units of a certain item in a, b, c and d hours respectively, where a, b, c, d are consecutive integers in increasing order. If A and C together can produce 60 units in more than 2 hours, then which of the following can be the time they will take to produce 60 units, if all are working together?

A. 28.2 minutes
B. 46.75 minutes
C. 60 minutes
D. 63.15 minutes
E. 79 minutes

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Originally posted by EgmatQuantExpert on 12 Apr 2019, 03:56.
Last edited by chetan2u on 18 Apr 2019, 02:19, edited 1 time in total.
Corrected the question
Math Expert V
Joined: 02 Aug 2009
Posts: 7957
Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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EgmatQuantExpert wrote:
Question of the Week #43

While working individually, each of A, B, C and D can produce 60 units of a certain item in a, b, c and d hours respectively, where a, b, c, d are consecutive integers in increasing order. If A and C together can produce 60 units in more than 2 hours, then which of the following cannot be the time they will take to produce 60 units, if all are working together?

A. 28.2 minutes
B. 46.75 minutes
C. 60 minutes
D. 63.15 minutes
E. 79 minutes

so a, b, c and d can be written as a, a+1, a+2 and a+3...

If A and C together can produce 60 units in more than 2 hours...$$\frac{1}{a}+\frac{1}{c}<\frac{1}{2}$$...
Just substitute a..
a as 1.. c is 3.. $$1+(\frac{1}{3})=\frac{4}{3} >\frac{1}{2}$$
a as 3.. c is 5...$$(\frac{1}{3})+(\frac{1}{5})=\frac{8}{15}>\frac{1}{2}$$
So, minimum value of a is 4..

Thus the least time all four will take is if a, b, c, and d are 4, 5, 6 and 7...
$$\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}=\frac{5*6*7+4*6*7+4*5*7+4*5*6}{4*5*6*7}=\frac{210+168+140+120}{840}=\frac{638}{840}.......$$..
so time = 840/638 hrs = $$\frac{840*60}{638}$$ minutes = 78.99 minutes..
So, the least value is 78.99..
Thus all choices A to D are not possible..

E

EgmatQuantExpert, please relook what are you looking for.. which of the following cannot or can be the time they will take to produce 60 units
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Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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chetan2u wrote:
EgmatQuantExpert wrote:
Question of the Week #43

While working individually, each of A, B, C and D can produce 60 units of a certain item in a, b, c and d hours respectively, where a, b, c, d are consecutive integers in increasing order. If A and C together can produce 60 units in more than 2 hours, then which of the following cannot be the time they will take to produce 60 units, if all are working together?

A. 28.2 minutes
B. 46.75 minutes
C. 60 minutes
D. 63.15 minutes
E. 79 minutes

so a, b, c and d can be written as a, a+1, a+2 and a+3...

If A and C together can produce 60 units in more than 2 hours...$$\frac{1}{a}+\frac{1}{c}<\frac{1}{2}$$...
Just substitute a..
a as 1.. c is 3.. $$1+(\frac{1}{3})=\frac{4}{3} >\frac{1}{2}$$
a as 3.. c is 5...$$(\frac{1}{3})+(\frac{1}{5})=\frac{8}{15}>\frac{1}{2}$$
So, minimum value of a is 4..

Thus the least time all four will take is if a, b, c, and d are 4, 5, 6 and 7...
$$\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}=\frac{5*6*7+4*6*7+4*5*7+4*5*6}{4*5*6*7}=\frac{210+168+140+120}{840}=\frac{638}{840}.......$$..
so time = 840/638 hrs = $$\frac{840*60}{638}$$ minutes = 78.99 minutes..
So, the least value is 78.99..
Thus all choices A to D are not possible..

E

EgmatQuantExpert, please relook what are you looking for.. which of the following cannot or can be the time they will take to produce 60 units

Is there a typo in the question? It says which of the following "cannot" be the time taken.

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Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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given a,b,c,d are consective integers
so a=x ,b=x+1, c= x+2, d= x+3
also given
1/a + 1/c>1/2
use hit and trial to see at which value of x we get 1/a + 1/c>1/2
only at a=4 and c=6 we get >1/2
so a=4,b=5,c=6,d=7
working together
1/4+1/5+1/6+1/7 = 5*6*7+4*6*7+4*5*7+4*5*6/ (4*5*6*7 )
solve we get
638/840 ; time 840*60/638 = ~79 mins
IMO E

EgmatQuantExpert wrote:
Question of the Week #43

While working individually, each of A, B, C and D can produce 60 units of a certain item in a, b, c and d hours respectively, where a, b, c, d are consecutive integers in increasing order. If A and C together can produce 60 units in more than 2 hours, then which of the following cannot be the time they will take to produce 60 units, if all are working together?

A. 28.2 minutes
B. 46.75 minutes
C. 60 minutes
D. 63.15 minutes
E. 79 minutes

Intern  B
Joined: 23 Mar 2013
Posts: 4
Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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1
a + c > 2 and a<b<c<d,
means, a will take at least 4 hours to complete the work
in an hour, units of work completed,

for A, 60*1/4 = 15
B, 60*1/5 = 12
C, 60*1/6=10
D, 60*1/7 ~ 8

Total work = 15+12+10+8 = 45
remaining work= 60-45= 15 which 4th of total, so would need additional 4th of an hour, i.e. 15mins,

in all the minimum time would be greater than, 60 + 15 = 75, the only answer which is greater than 75, is E

Originally posted by vijay282001 on 12 Apr 2019, 18:56.
Last edited by vijay282001 on 13 Apr 2019, 05:08, edited 1 time in total.
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Posts: 3344
Location: India
GPA: 3.12
Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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EgmatQuantExpert wrote:
While working individually, each of A, B, C and D can produce 60 units of a certain item in a, b, c and d hours respectively, where a, b, c, d are consecutive integers in increasing order. If A and C together can produce 60 units in more than 2 hours, then which of the following cannot be the time they will take to produce 60 units, if all are working together?

A. 28.2 minutes
B. 46.75 minutes
C. 60 minutes
D. 63.15 minutes
E. 79 minutes

For work to complete in 2 hours or less, A and C must complete half the work in an hour. The combined rate
of work of A and C cannot be greater than 30 units in an hour. B and D will complete more than 30 units/hr.
With increasing time, the rate at which they individually complete the work decreases.

Let us assume that A completes the work at 15 units/hr. A will take 4 hours ($$\frac{60}{15} = 4$$)

Since the time at which they individually complete work is in consecutive order,
B will take 5 hours and the work done by A in an hour is $$\frac{60}{5} = 12$$ units.
C will take 6 hours and the work done by A in an hour is $$\frac{60}{6} = 10$$ units.
D will take 7 hours and the work done by A in an hour is $$\frac{60}{7} = 8.5$$ units. (approximately)
In an hour, they will complete a total of $$15 + 12 + 10 + 8.5 = 45.5$$ units.

Therefore, the time take to complete the work is $$\frac{60}{45.5} = 1.32$$ hours (or) 79 minutes(Option E)
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Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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Because of the structure of the question, we can devise a shortcut to solving this problem – one that takes only a few seconds - by employing a bit of logic:

In solving questions like this we have to fix a minimum or a maximum time for the value to be determined (in this case, the time that all four will take to complete the job). In this particular case, however, we can only fix a minimum time because there is no maximum limit set: A & C could take a 100 or even a million hours to complete the job. It follows that when A, B, C and D all work together there is no maximum time limit and we can only fix a minimum value since there is a fixed relationship between the times that each take to finish the job while working alone.
It stands to reason that the minimum time in which A, B, C and D can complete the work must lie between the highest value (E) and the second highest value (D) thus making E the only possible answer because, if the minimum time is less (i.e. between any two of the lesser answer choices), all the answer choices which are more than the minimum time become possible. For example, if the minimum time is 50 minutes (i.e. between B and C), that would make C, D and E all possible but the question allows for only one correct answer. If the minimum time was more than 79 minutes (E) then the correct answer would have been "None" which is not among the answer choices. So the correct answer can only be E. On the other hand, if the minimum time is less than (A), all the options (A) through (E) could be possible and the correct answer would be "All" which is also not among the given options.
This shortcut could be prevented by changing the structure of the question to allow for more than one correct answer (e.g. D and E, A only, E only, all five, none, etc.).

Originally posted by effatara on 18 Apr 2019, 02:17.
Last edited by effatara on 22 Apr 2019, 01:16, edited 2 times in total.
Intern  Joined: 05 Feb 2014
Posts: 9
Re: Question of the Week - 43 (While working individually, each of A, B)  [#permalink]

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Hi everyone,

I'm a bit confused and I was hoping to get some clarification on this.

I see an answer that states:
(1/a + a/c) > 1/2
Why does a has to be at least 4? If a were to be 1, (1/1 + 1/3) is indeed greater than 1/2.

Another answer states the opposite:
(1/a + 1/c) < 1/2
In this case, I do understand that a must be at least 4, but why are we posing (1/a + 1/c) less than 1/2 when we know that it takes a+c at least 2 hours to build 60 units?

Thanks so much for helping!
Mike Re: Question of the Week - 43 (While working individually, each of A, B)   [#permalink] 20 Apr 2019, 21:37
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