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Three men and eight machines can finish a job in half the time taken

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Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 02:27
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Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?

A. 10
B. 11
C. 12
D. 13
E. 14

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Re: Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 02:49
rate of one machine ; 13*2 ; 26 days
and rate of man ; x
3/x+8/26 = 2*(8/x+3/26)
solve for x
we get x= 169
so in 13 days ; 169/13 ; 13 men would be required
IMO D


Bunuel wrote:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?

A. 10
B. 11
C. 12
D. 13
E. 14

Project PS Butler


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Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 02:50
1
time taken by 3Men + 8 Machines = half * time taken by 3 machines + 8 men
In other words time taken by 3 men + 8 machines = time taken by 6 machines + 16 men
Two machines can finish the work in 13 days means that one machine can finish the work in 26 days.

let the time take for a man to finish the work be x, then

3/x + 8/26 = 6/26 + 16/x
2/26 = 13/x
x = 13*13
Hence 13 men will take 13*13/13 = 13 days.

The answer is D.
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Re: Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 03:01
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Bunuel wrote:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?

A. 10
B. 11
C. 12
D. 13
E. 14



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Answer: Option D


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Re: Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 03:33
1
Bunuel wrote:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?

A. 10
B. 11
C. 12
D. 13
E. 14

Project PS Butler


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Solution:


Let A and B be the per day work of one man and one machine respectively
    • Work done by 3 men and 8 machines in one day = 3A + 8B
    • Work done by 8 men and 3 machines in one day = 8A + 3B
      o Time taken to complete the work by 3 men and 8 machines is half time by 8 men and 3 machines
      o Work done by 3 men and 8 machines in one day is twice of the work done by 8 men and 3 machines in one day
      o \(3A + 8B = 2(8A + 3B)\)
      o \(3A +8B = 16A + 6B\)
      o \(8B-6B = 16A – 3A\)
      o \(2B = 13A\)
    • Work done by 2 machines in one day is equal to the work done by 13 men in one day.
    • 2 machines can complete the job in 13 days
      o 13 men can complete the job in 13 days
Hence, the correct answer is Option D.
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Re: Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 04:17
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Since, TEAM-1 (3 men and 8 machines) can finish a job in half the time taken by TEAM-2 (3 machines and 8 men) to finish the same job, we can express the TWO TEAM'S CAPACITY through the following equation:
3 MEN+ 8 MACHINES= 2 (3 MACHINES+ 8 MEN)
=> 3 MEN+8 MACHINES= 6 MACHINES+16 MEN
=>8 MACHINES- 6 MACHINES= 16 MEN- 3 MEN
=> 2 MACHINES=13 MEN
Therefore, we can infer that THE CAPACITY OF TWO MACHINES IS EQUIVALENT TO THAT OF 13 MEN.
Hence, it requires 13 men to complete what can be done by 2 machines in the same amount of time.

The answer is D.

Bunuel, please convey me whether my approach is efficient.
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Re: Three men and eight machines can finish a job in half the time taken  [#permalink]

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New post 31 Mar 2020, 09:20
1
Bunuel wrote:
Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days?

A. 10
B. 11
C. 12
D. 13
E. 14


two machines can finish the job in 13 days
--> 1 machine can complete a job in 26 days
Let 1 man complete a job in '\(m\)' days

Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job
--> \(\frac{3}{m} + \frac{8}{26} = 2(\frac{3}{26} + \frac{8}{m})\)
--> \(\frac{8}{26} - \frac{6}{26} = \frac{16}{m} - \frac{3}{m}\)
--> \(\frac{1}{13} = \frac{13}{m}\)
--> \(m = 169\) days

Number of men required to finish the job in 13 days = \(\frac{169}{13} = 13\) men

Option D
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Re: Three men and eight machines can finish a job in half the time taken   [#permalink] 31 Mar 2020, 09:20

Three men and eight machines can finish a job in half the time taken

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