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

[AbhishekP]

A bath tub can be filled by the cold water and hot water pipes in 10 mins and 15 mins, respectively. A person leaves the bathroom after turning on both pipes simultaneously and returns at the moment when the bath should be full. Finding, however, that the waste pipe has been open, he then closes it. In exactly 4 mins more the bath is fill. In how much time would the waste pipe empty the full bath, if it alone is opened.

The answer given to the above question is 15 mins and not 9 mins. How?

[JARED]

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