Question

Working in pairs, PQ, QR and RP can complete a job in 24 days, 20 days and 30 days respectively. Find the respective times taken by P, Q and R individually to complete the same job in days.

Answer 1

Solution :-

P and Q can complete a job together in 24 days

The, part of the job that P and Q can do in 1 day = 1/24

Q and R can complete a job together in 20 days

The, part of job that Q and R can do in 1 day = 1/20 

R and P can complete a job together in 30 days

Then, part of job that R and P can do in 1 day = 1/30

Let the time taken by P, Q and R to complete the job be p days, q days and r days respectively.

⇒ 1/p + 1/q = 1/24 ............(1)

⇒ 1/q + 1/r = 1/20 .............(2)

⇒ 1/r + 1/p = 1/30 .............(3)

Adding (1) and (2), we get

⇒ 1/p + 1/r + 2/q = 1/24 + 1/20

Now, subtracting (3) from it, we get

1/p + 1/r are cancelled.

⇒ 2/q = 1/24 + 1/20 - 1/30

Taking LCM of 24, 20 and 30 and then solving it.

1/24 + 1/20 - 1/30

(5 + 6 - 4)/120

= (11- 4)/120

= 7/120

⇒ 2/q = 7/120

⇒ 7q = 2*120

⇒ q = 240/7 days

Putting the value of q = 240/7 in (1)

1/p + 1/q = 1/24

⇒ 1/p + 7/240 = 1/24

⇒ 1/p = 1/24 - 7/240

1/p = (10 - 7)/240

⇒ 1/p = 3/240

⇒ p = 240/3

⇒ p = 80 days

Putting the value of p = 80 in (3)

⇒ 1/r + 1/p = 1/30

⇒ 1/r + 1/80 = 1/30

⇒ 1/r = 1/30 - 1/80

⇒ 1/r = (8 - 3)/240

⇒ 1/r = 5/240

⇒ r = 240/5

⇒ r = 48 days

So, P, Q and R individually take 80 days, 240/7 days and 48 days respectively to complete the same job.