(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?
(In reply to some thoughts on the problem
by K Sengupta)
Let H, C and P respectively denote the hot water tap, the cold
water tap and the plug. By the problem, each of H and C fills
one-sixth of the bath in i minute while P empties one-fourth
of the bath in the same time.
Accordingly, with H and C running simultaneously, precisely
1/3rd of the bath will be filled in 1 minute.
Therefore, the combination of H, C and P will fill
(1/6+1/6-1/4) = 1/12th of the bath each minute, while
the combination of C and P will empty 1/4 - 1/6 = 1/12th of
the bath in 1 minute.
Since the combination of H, C and P can run for 3 minutes,
precisely 3/12 = 1/4th of the bath will be filled by the
stipulated time. After the stipulated time (3 minutes),
the combination of C and P will empty (5/12)*(1/2) = 5/24th
of the bath in 2.5 minutes.
Thus, precisely 1/4 - 5/24 = 1/24th of the bath will fill up
in every 5.5 minutes. Accordingly, 3/4th of the bath will fill up in
(11/2)*(3/4)*(24) = 99 minutes.
The remaining 1/4th portion will fill up by the H,C, P combination
in a further 3 minutes.
Consequently, the required time taken to fill up the entire bath is
99+3 = 102 minutes; or 1 hour and 42 minutes.