Question

Pipes P & Q fill a cistern in 12 minutes when operated together.Q & R fill it in 20 minutes when operated together and P & R fill it in 15 minutes when operated together. In how many minutes can all 3 pipes fill the cistern if operated together?​

Answer 1

Solutions :-

Given :
Pipes P & Q fill a cistern in 12 minutes when operated together.
(P + Q)'s one minute work = 1/12

Q & R fill it in 20 minutes when operated together.
(Q + R)'s one minute work = 1/20

And P & R fill it in 15 minutes when operated together.
(P + R)'s one minute work = 1/15


Find the (P + Q + R)'s one minute work :-

(P + Q) + (Q + R) + (P + R) = 1/12 + 1/20 + 1/15
=> P + Q + Q + R + P + R = (5 + 3 + 4)/60
=> 2(P + Q + R) = 12/60
=> P + Q + R = 12/(60 × 2) = 1/10


Therefore,
(P + Q + R)'s one minute work = 1/10

Hence,
All the three pipes together can fill the cistern in 10 minutes.

Answer 2

P and Q fill a cistern in 12 min

Filled by both of them (P and Q) in 1 min = $\dfrac{1}{12}$ _(1)

Q and R fill it in 20 min

Filled by both (Q and R) in 1 min = $\dfrac{1}{20}$ _(2)

P and R fill it in 15 min

Filled by both (P and R) of them in 1 min = $\dfrac{1}{15}$ _(3)

We have to find how much time 3 pipes together take to fill a cistern.

Means we have to find P + Q + R

Add eq. (1), (2) and (3)

(P + Q) + (Q + R) + (P + R) = $\dfrac{1}{12}$ + $\dfrac{1}{20}$ + $\dfrac{1}{15}$

2 (P + Q + R) = $\dfrac{5\:+\:3\:+\:4}{60}$

P + Q + R = $\dfrac{12}{60}$ × $\dfrac{1}{2}$

P + Q + R = $\dfrac{1}{10}$

3 of pipes together filled that cistern in 10 min.