Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:

Part filled by (A + B + C) in 3 minutes = 3	❨	1	+	1	+	1	❩	=	❨	3 x	11	❩	=	11	.
		30		20		10					60			20

Part filled by C in 3 minutes =	3	.
	10

∴ Required ratio =	❨	3	x	20	❩	=	6	.
		10		11			11

Let B be turned off after x minutes. Then,

Part filled by (A + B) in x min. + Part filled by A in (30 -x) min. = 1.

∴ x	❨	2	+	1	❩	+ (30 - x).	2	= 1
		75		45			75

⟹	11x	+	(60 -2x)	= 1
	225		75

⟹ 11x + 180 - 6x = 225.

⟹ x = 9.

(A + B)'s 1 hour's work =	❨	1	+	1	❩	=	9	=	3	.
		12		15			60		20

(A + C)'s hour's work =	❨	1	+	1	❩	=	8	=	2	.
		12		20			60		15

Part filled in 2 hrs =	❨	3	+	2	❩	=	17	.
		20		15			60

Part filled in 6 hrs =	❨	3 x	17	❩	=	17	.
			60			20

Remaining part =	❨	1 -	17	❩	=	3	.
			20			20

Now, it is the turn of A and B and	3	part is filled by A and B in 1 hour.
	20

∴ Total time taken to fill the tank = (6 + 1) hrs = 7 hrs.

Part filled by A in 1 min =	1	.
	20

Part filled by B in 1 min =	1	.
	30

Part filled by (A + B) in 1 min =	❨	1	+	1	❩	=	1	.
		20		30			12

∴ Both pipes can fill the tank in 12 minutes.

Suppose, first pipe alone takes x hours to fill the tank .

Then, second and third pipes will take (x -5) and (x - 9) hours respectively to fill the tank.

∴	1	+	1	=	1
	x		(x - 5)		(x - 9)

⟹	x - 5 + x	=	1
	x(x - 5)		(x - 9)

⟹ (2x - 5)(x - 9) = x(x - 5)

⟹ x² - 18x + 45 = 0

(x - 15)(x - 3) = 0

⟹ x = 15.    [neglecting x = 3]

Time taken by one tap to fill half of the tank = 3 hrs.

Part filled by the four taps in 1 hour =	❨	4 x	1	❩	=	2	.
			6			3

Remaining part =	❨	1 -	1	❩	=	1	.
			2			2

∴	2	:	1	:: 1 : x
	3		2

⟹ x =	❨	1	x 1 x	3	❩	=	3	hours i.e., 45 mins.
		2		2			4

So, total time taken = 3 hrs. 45 mins.

Net part filled in 1 hour	❨	1	+	1	-	1	❩	=	17	.
		5		6		12			60

∴ The tank will be full in	60	hours i.e., 3	9	hours.
	17		17

Work done by the leak in 1 hour =	❨	1	-	3	❩	=	1	.
		2		7			14

∴ Leak will empty the tank in 14 hrs.

Part filled in 4 minutes = 4	❨	1	+	1	❩	=	7	.
		15		20			15

Remaining part =	❨	1 -	7	❩	=	8	.
			15			15

Part filled by B in 1 minute =	1
	20

∴	1	:	8	:: 1 : x
	20		15

x =	❨	8	x 1 x 20	❩	= 10	2	min = 10 min. 40 sec.
		15				3

∴ The tank will be full in (4 min. + 10 min. + 40 sec.) = 14 min. 40 sec.

Part filled by (A + B) in 1 minute =	❨	1	+	1	❩	=	1	.
		60		40			24

Suppose the tank is filled in x minutes.

Then,	x	❨	1	+	1	❩	= 1
	2		24		40

⟹	x	x	1	= 1
	2		15

⟹ x = 30 min.