P, Q and R are working on a project. Q alone would have been able to

Question

P, Q and R are working on a project. Q alone would have been able to complete the project in two hours less than R would. If Q and R start working and Q leaves after three hours, then R would need one hour more to finish the work. If P and Q start working and P leaves after two hours, then Q would need one hour more to finish the work. If all three of them together complete the work in less than 2 hours, then the ratio of the time taken by R alone to finish the work to that taken by P alone.

A. 8:5
B. 7:4
C. 3:1
D. 2:1
E. 1:2

Source: CAT Question adopted to GMAT format

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Official Answer

D

A. 8:5
B. 7:4
C. 3:1
D. 2:1
E. 1:2

Source: CAT Question adopted to GMAT format

23ss23 · Answer

Q)  P, Q and R are working on a project. Q alone would have been able to complete the project in two hours less than R would. If Q and R start working and Q leaves after three hours, then R would need one hour more to finish the work. If P and Q start working and P leaves after two hours, then Q would need one hour more to finish the work. If all three of them together complete the work in less than 2 hours, then the ratio of the time taken by R alone to finish the work to that taken by P alone.

Ans. The question may seem complicated at first, but it's actually quite straightforward if you break it down carefully. Let's go step by step to simplify it -

From the first scenario: "If Q and R begin working together and Q leaves after three hours, R will require one additional hour to complete the task," we can infer that the total work equals 3 hours of Q's effort and 4 hours of R's effort.

In the second scenario: "If P and Q start working together and P leaves after two hours, Q will need one more hour to finish the task," we deduce that the total work is equivalent to 2 hours of P's effort and 3 hours of Q's effort.

By combining these two scenarios, we see that Q contributes 3 hours of work in both cases. This leads us to conclude that 4 hours of R's work is equivalent to 2 hours of P's work. Consequently, the ratio of the time required by R and P to finish the work is 2:1.

nishantswaft · Answer

Interesting question though detail bulky but once we start forming equation it gets easier to solve.

If Q and R begin working together and Q leaves after three hours, R will require one additional hour to complete the task i.e. W=(Q+R)*3+R*1=3Q+4R

If P and Q start working together and P leaves after two hours, Q will need one more hour to finish the task i.e. W=(P+Q)*2+Q*1

Equate both equations we'll get,
2P=4R and P=2R

Now also since work=rate *time
R*time by R=P*time by P
Or our ratio will be 2:1

Kinshook · Answer

P, Q and R are working on a project. Q alone would have been able to complete the project in two hours less than R would. If Q and R start working and Q leaves after three hours, then R would need one hour more to finish the work. If P and Q start working and P leaves after two hours, then Q would need one hour more to finish the work. If all three of them together complete the work in less than 2 hours, then the ratio of the time taken by R alone to finish the work to that taken by P alone.

A. 8:5
B. 7:4
C. 3:1
D. 2:1
E. 4:3

If all three of them together complete the work in less than 2 hours, then the ratio of the time taken by R alone to finish the work to that taken by P alone.

Let the time taken by P, Q & R working alone to complete the project be p, q & r hours respectively

Q alone would have been able to complete the project in two hours less than R would.
q = r - 2;  r = q + 2

If Q and R start working and Q leaves after three hours, then R would need one hour more to finish the work.
3/q + 3/r + 1/r = 1
3/q + 4/(q+2) = 1
3(q+2) + 4q = q(q+2)
7q + 6 = q^2 + 2q
q^2 - 5q - 6 = 0
(q+1)(q-6) = 0
 q = 6 
 r = q + 2 = 8

If P and Q start working and P leaves after two hours, then Q would need one hour more to finish the work .  
2/p + 2/q + 1/q = 1
2/p + 3/6 = 1
2/p = 1/2
 p = 4

If all three of them together complete the work in less than 2 hours, then the ratio of the time taken by R alone to finish the work to that taken by P alone.
1/p + 1/q + 1/r = 1/4 + 1/6 + 1/8 = (6+4+3)/24 = 13/24
Number of hours required = 24/13 < 2 hours

The ratio of the time taken by R alone to finish the work to that taken by P alone = r:p = 8:4 = 2:1

IMO D

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