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P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours

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P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours  [#permalink]

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New post 13 Feb 2019, 06:17
1
6
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

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P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours, while working individually. To finish the job, they started working together in pairs. However, none of the pairs work for more than an hour at a stretch and none of the persons can work for more than 2 hours at a stretch. What is the least time by which the job will be finished ?

    A. 4 hours
    B. 5 hours
    C. 6 hours
    D. 8 hours
    E. 12 hours


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Re: P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours  [#permalink]

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New post 13 Feb 2019, 06:42
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Total work to be done = LCM(10,12,15) = 60 Units.

P can do 6 units per hour.
Q can do 5 units per hour.
R can do 4 units per hour.

Consider PQ+QR+RP.

11+9+10 in 3 hours.
30 units in 3 hours.
So 60 units in 6 hours.

C is the answer

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P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours  [#permalink]

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New post Updated on: 14 Feb 2019, 09:34
rate
p=1/10 ,q=1/12 r= 1/15
pairs they can work together
PR= 1/10 +1/15 = 10/60
PQ = 1/10 + 1/12= 11/60
QR = 1/12 + 1/15 = 9/60

so together thwy can work in
10/60+11/60+9/60 = 1/2
so half work done in 3 hrs ; so 1 complete work would be done in 6 hrs
IMO C


EgmatQuantExpert wrote:
P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours, while working individually. To finish the job, they started working together in pairs. However, none of the pairs work for more than an hour at a stretch and none of the persons can work for more than 2 hours at a stretch. What is the least time by which the job will be finished ?

    A. 4 hours
    B. 5 hours
    C. 6 hours
    D. 8 hours
    E. 12 hours


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Originally posted by Archit3110 on 13 Feb 2019, 07:37.
Last edited by Archit3110 on 14 Feb 2019, 09:34, edited 1 time in total.
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Re: P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours  [#permalink]

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New post 13 Feb 2019, 22:59
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From a group of three people, pairs have to be formed. This can only be done in 3 ways. PQ, PR, QR.

So in this question, we have to solve for the least time taken by using only these 3 pairs.

Times taken by each and times taken by each pair are:

P = \(\frac{1}{10}\)

Q = \(\frac{1}{12}\)

R = \(\frac{1}{15}\)

PQ = \(\frac{1}{10}\) + \(\frac{1}{12}\) = \(\frac{11}{60}\)

QR = \(\frac{1}{12}\) + \(\frac{1}{15}\) = \(\frac{9}{60}\)

PR = \(\frac{1}{10}\) + \(\frac{1}{15}\) = \(\frac{10}{60}\)

In order to find the least possible time to complete the work, we have to use a couple who can do maximum work in an hour. So out of 3 pairs above, arranging the teams in decreasing order of their ability to do work in an hour: PQ>PR>QR.

So PR is the couple who can complete maximum work in an hour and QR is the couple with the least amount of work done in an hour.

So let's start the work with PQ now.

1st hour: PQ

Now PQ together cannot work in the second hour, so out of PR and QR, any team can work. As PR completes next maximum possible work in an hour after PQ we will consider PR to work in 2nd hour.

2nd hour: PR

Now PR cannot work together in the third hour. And also P cannot work in the third hour as it already worked for 2 continuous hours. So the remaining choice is only QR.

3rd hour: QR

So work done in 3 hours is \(\frac{11}{60}\) + \(\frac{10}{60}\) + \(\frac{9}{60}\) = \(\frac{30}{60}\)

Half of the work is done in 3 hours.

Now in fourth hour R cannot work as R already worked for 2 continuous hours. So the only possible choice is PQ.

4th hour: PQ

Now in fifth hour Q cannot work as Q already worked for 2 continuous hours. So the only possible choice is PR.

5th hour: PR

Now in sixth hour P cannot work as P already worked for 2 continuous hours. So the only possible choice is QR.

6th hour: QR

Work done in 4th, 5th and 6th hours is \(\frac{11}{60}\) + \(\frac{10}{60}\) + \(\frac{9}{60}\) = \(\frac{30}{60}\)

Remaining Half of the work is done in another 3 hours.

So the least possible time to complete the work is 6 hours.

OPTION: C
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Re: P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours  [#permalink]

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New post 14 Feb 2019, 22:32

Solution



Given:
In this question, we are given
    • P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours, while working individually.
    • To finish the job, they started working together in pairs.
    • None of the pairs work for more than an hour at a stretch.
    • None of the persons can work for more than 2 hours at a stretch.

To find:
We need to determine
    • The least time by which the job will be finished.

Approach and Working:
Let us assume that the total job = LCM (10, 12, 15) = 60 units

Hence, in 1 hour,
    • P can do = \(\frac{60}{10}\) = 6 units
    • Q can do = \(\frac{60}{12}\) = 5 units
    • R can do = \(\frac{60}{15}\) = 4 units

As they are working in pairs, as a pair
    • 1-hour job of (P and Q) = (6 + 5) units = 11 units
    • 1-hour job of (P and R) = (6 + 4) units = 10 units
    • 1-hour job of (Q and R) = (5 + 4) units = 9 units

Therefore, in a span of 3 hours, the total work gets completed = (11 + 10 + 9) units = 30 units

So, to complete the total work, they will take another 3 hours or total 6 hours.

Hence the correct answer is Option C.

Answer: C

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Re: P, Q and R can do a certain job in 10 hours, 12 hours and 15 hours   [#permalink] 14 Feb 2019, 22:32
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