How to solve this riddle. Please do write the steps?

We have three rates:

h: rate of hot water in volume per minute
c: rate of cold water in volume per minute
d: rate at which waster pipe empties in volume per minute

The volume of the bath tub is unknown, and we'll call that V (and we will see that this value will cancel out in the equations--so we don't need to know it).

Tub can be filled by cold pipe in 10 mins: 10c = V --> c = V/10
Tub can be filled by hot pipe in 15 mins: 15h = V --> h = V/15

Initially the water is coming in at a rate of c + h (because both pipes are open), BUT the drain is also open, so we subtract off that rate, to get:

c + h - d = V/10 + V/15 - d

We leave it going for the amount of time it takes to fill the tub if the drain were not open...find this time:

(c + h)T = V --> (V/10 + V/15) * T = V
--> the V's cancel, leaving

T = 1 / (1/10 + 1/15) = 30 / (3 + 2) = 30/5 = 6 minutes

However, after this 6 minutes we find the water in the tub isn't full and once we close the drain, it takes four more minutes to fill up. Find the volume (as a fraction of V, the total volume) in the tub after this 6 minutes:

(c + h - d) * 6 = xV <-- x is some (positive) fraction of the total volume

We now close the drain, and find it takes 4 minutes to fill up the tub the rest of the way.

(c + h) * 4 = V - xV = (1 - x)V
--> again, the V's cancel and we can now solve for x

4(1/10 + 1/15) = (1 - x) --> x = 1 - 4/10 - 4/15 = 6/10 - 4/15 = (18 - 8)/30 = 10/30 = 1/3

This means that after the 6 minutes the bath tub was 1/3 full. This is not surprising, if it takes 6 minutes to fill the tub, but after the 6 minutes, it takes another 4, this means that you needed 2/3 of the full time, so the tub must have been 1/3 full.

Now that we know x, we can use the equation with d, to find d.

(c + h - d) * 6 = V/3
-->

c + h - d = V/18 --> d = c + h - V/18 = V/10 + V/15 - V/18
-->

d = V * (1/10 + 1/15 - 1/18) = V * (9/90 + 6/90 - 5/90) = V * 10/90 = V/9

Now find the time it takes d to drain the full tub (the full volume V):

d * T = V
--> the V's cancel, leaving

T = 1/d = 9 minutes