(This is an old chestnut of a puzzle, but with a slight twist)

It takes six minutes for just the cold tap to fill my bath with water.

The hot tap fills the bath at the same rate, but unfortunately (due to some strange plumbing) after the hot tap has been running for three minutes its supply of water shuts off for a full two and a half while the hot water tank refills and reheats - after the two and a half minutes, the hot tap commences dispensing water once again.

With the plug pulled out, a full bath empties (at a uniform rate) in only four minutes.

With the hot water tank full and the plug pulled out, how long will it take my bath to fill if I turn both taps on?

For the first 3 minutes, both taps are running and the plug is pulled out. So the rate the tub is being filled is: (bpm = bathtubs per minute)
1/3 bpm - 1/4 bpm = 1/12 bpm
Rb = 1/12 bpm (Rate that bathtub is filled when both taps are on)
Tb = 3 minutes (time that both taps are on)

then after that, for the next 2.5 minutes, only one tap is running, so the second rate is:
Rc = 1/6 bpm - 1/4 bpm = -1/12 bpm
Tc = 2.5 minutes

formula for how much the bathtub is filled after a certain amount of time is Rate X Time.

So after the first 3 minutes the bathtub is 1/12 bpm * 3 min = 1/4 bathtub filled. for reasons shown later, we'll make this 6/24 filled

Then after the next 2.5 minutes, the bathtub drains by -1/12 bpm * 2.5 minutes = -5/24 bathtubs.

to visualize this better:
+3.0 min = 6/24 filled (1/12 filled)
+2.5 min = 1/24 filled
+3.0 min = 7/24 filled
+2.5 min = 2/24 filled

so after every interval, it goes up by 6/24, then down by 5/24, then up again.
how long will it take before we get to 24/24, or 1 whole bathtub filled?

let
x = number of times that both taps are on
y = number of times that cold tap is on

x(6/24) - y(5/24) = 24/24
multiply both sides by 24:

6x - 5y = 24

as levik pointed out, once we finally fill the tub, we stop the time. So we have one more x than y, or algebraically y = x - 1

so:
6x - 5(x-1) = 24
x = 19

and then
y = 18

but we're not done here.
x and y are just the number of times that both taps and just the cold tap is on (respectively).

total time that both taps are on: 3 min * 19 intervals = 57 minutes

total time that cold tap is on: 2.5 min * 18 intervals = 45 minutes

total time to fill the tub: 57 + 45 = 102 minutes

Posted by Happy on 2002-07-03 06:09:03