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01 Jul 2025, 11:58
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First, let's determine the number of days the machines worked in pairs. This is the number of unique pairs you can form from 20 machines, which is a combination calculation "20 choose 2":
Number of pairs (days) = C(20,2)=
=190 days.
Now, let the sum of the rates of all 20 machines be R.
R=r1+r2+...+r20
In the 190 days of pair work, each machine is paired with the other 19 machines. This means each machine's rate contributes to the work on 19 different days. So, the total work done in this phase is:
Work by pairs=19R.
Calculate the Work Done Together
After the pair work, all 20 machines work together for 6 additional days. The combined rate is R.
Work done together = 6×R=6R.
Find the Total Work for the Job
The total work to complete the job is the sum of the work from both phases.
Total Work = (Work by pairs) + (Work done together)
Total Work = 19R+6R=25R.
The question asks how long it would take if all 20 machines, with their combined rate R, worked together from the start.
Time = Total Work / Combined Rate
Time = 25R/R
Time = 25 days
Number of pairs (days) = C(20,2)=
=190 days.
Now, let the sum of the rates of all 20 machines be R.
R=r1+r2+...+r20
In the 190 days of pair work, each machine is paired with the other 19 machines. This means each machine's rate contributes to the work on 19 different days. So, the total work done in this phase is:
Work by pairs=19R.
Calculate the Work Done Together
After the pair work, all 20 machines work together for 6 additional days. The combined rate is R.
Work done together = 6×R=6R.
Find the Total Work for the Job
The total work to complete the job is the sum of the work from both phases.
Total Work = (Work by pairs) + (Work done together)
Total Work = 19R+6R=25R.
The question asks how long it would take if all 20 machines, with their combined rate R, worked together from the start.
Time = Total Work / Combined Rate
Time = 25R/R
Time = 25 days
01 Jul 2025, 12:12
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There are a total of 20 machines and on each day, one unique pair works together. Total possible unique pairs are 20C2. That amounts to 190 ways.
However, looking at the problem in this manner can be confusing. Instead look at it this way:
With machine 1, 20 combinations are possible.
Which machine 2, 20 combinations are possible, and so on.
Presume on day 1, M1 and M2 worked together. Let their work rates be R1 and R2 respectively.
On day 1, work done is 1 * (R1 + R2). Since 20 combinations will be made with R1, in the final sum of work done on days before the 6 days when all machines worked together, r1 would have been added 20 times.
Same goes for other machines. Essentially, work done to complete the job is 20*(R1 + R2 +...R20) + 6*(R1+R2+....R20).
If all machines were to work together from the start, they would take 20+6 = 26 days.
Hence the correct answer is option C.
However, looking at the problem in this manner can be confusing. Instead look at it this way:
With machine 1, 20 combinations are possible.
Which machine 2, 20 combinations are possible, and so on.
Presume on day 1, M1 and M2 worked together. Let their work rates be R1 and R2 respectively.
On day 1, work done is 1 * (R1 + R2). Since 20 combinations will be made with R1, in the final sum of work done on days before the 6 days when all machines worked together, r1 would have been added 20 times.
Same goes for other machines. Essentially, work done to complete the job is 20*(R1 + R2 +...R20) + 6*(R1+R2+....R20).
If all machines were to work together from the start, they would take 20+6 = 26 days.
Hence the correct answer is option C.
Bunuel
01 Jul 2025, 12:26
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20 machines, all different rates.
All machine work together at Rate/day = R (Lets say)
Exactly two machines are assigned to work each, no pair work more than once.
Each machine will work for 19 days, ( instance of working with every other machine, Total No of pair with Each),
Work done by these until all paired once = 19R
all work together for 6 days, work done= 6R
Total Work = W = 19R+ 6R = 25R
No of days required = W/R = 25
B is the answer
All machine work together at Rate/day = R (Lets say)
Exactly two machines are assigned to work each, no pair work more than once.
Each machine will work for 19 days, ( instance of working with every other machine, Total No of pair with Each),
Work done by these until all paired once = 19R
all work together for 6 days, work done= 6R
Total Work = W = 19R+ 6R = 25R
No of days required = W/R = 25
B is the answer
Bunuel
