Correct option is (b) 6 hours
First of all, let us assume that the first pipe fills the swimming pool in x hours.
The time taken by the first pipe to fill the pool = x hours ……..(1)
The part of the pool filled in a unit time by the first pipe = \(\frac 1x\) ………(2)
It is given that the second pipe fills the pool 5 hours faster than the first pipe.
From equation (1), we have the time taken by the first pipe to fill the pool.
The times taken by the second pipe to fill the pool = (x − 5) hours ……..(3)
The part of the pool filled in a unit time by the second pipe = \(\frac 1{(x - 5)}\) …...…(4)
It is given that the third pipe fills the pool 4 hours faster than the second pipe.
From equation (3), we have the time taken by the second pipe to fill the pool.
The times taken by the third pipe to fill the pool = (x − 5 − 4) = (x − 9) hours ……..(5)
The part of the pool filled in a unit time by the third pipe = \(\frac 1{(x -9)}\) ………(6)
It is also given that the first pipe and the second pipe operating simultaneously fill the pool at the same time as that taken by the third pipe alone. It means that the part of the pool filled in a unit time by the first pipe and the second pipe is equal to the part of the pool filled in a unit time by the third pipe.
From equation (2), equation (4), and equation (6), we have the part of the pool filled in a unit time by the first pipe, the second pipe, and the third pipe respectively. So,
So, x = 3 and x = 15.
From equation (3) and equation (5), we have the time taken by the first pipe and the second pipe to fill the pool.
If we take x = 3, then the time taken by the second pipe and the third pipe will be negative. So, x = 3 is not possible.
Now, putting the value of x in equation (1), equation (3), and equation (5), we get
The time taken by the first pipe to fill the pool = 15 hours.
The time taken by the second pipe to fill the pool = (15 − 5) hours = 10 hours.
The time taken by the third pipe to fill the pool = (15 − 9) hours = 6 hours.