Let the rate of the smaller pipe be x and the rate of the larger pipe be y (in units of pool volume per hour).

We know that together they can fill the pool in 12 hours, so:

x + y = 1/12

12x + 12y = 1 . . . . . (i)

We also know that if the smaller pipe is used for 4 hours, it fills up 4x of the pool volume. Similarly, if the larger pipe is used for 9 hours, it fills up 9y of the pool volume.

So, after 4 hours of the smaller pipe and 9 hours of the larger pipe, half of the pool is filled:

4x + 9y = 1/2

8x + 18y = 1 . . . . . (ii)

Multiplying the (i) by 2 and (ii) by 3, we get :

24x + 24y = 2 . . . . (iii)

24x + 54y = 3 . . . . (iv)

Subtracting (iii) from (iv)

30y = 1

y = 1/30,

Now, substituting y = 1/30 in (i), we get :

12x + 12(1/30) = 1

12x = 1 - 2/5

12x = 3/5

x = 1/20

To fill the entire pool:

The smaller pipe would take:

1 / (1/20) = 20 hours

The larger pipe would take:

1 / (1/30) = 30 hours