Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is:

(x x 5) = (0.75 x 8)   ⟹ x =	❨	6	❩	= 1.20
		5

Quantity of milk =	❨	60 x	2	❩litres = 40 litres.
			3

Quantity of water in it = (60- 40) litres = 20 litres.

New ratio = 1 : 2

Let quantity of water to be added further be x litres.

Then, milk : water =	❨	40	❩	.
		20 + x

Now,	❨	40	❩	=	1
		20 + x			2

⟹ 20 + x = 80

⟹ x = 60.

∴ Quantity of water to be added = 60 litres.

Let A = 2k, B = 3k and C = 5k.

A's new salary =	115	of 2k =	❨	115	x 2k	❩	=	23k
	100			100				10

B's new salary =	110	of 3k =	❨	110	x 3k	❩	=	33k
	100			100				10

C's new salary =	120	of 5k =	❨	120	x 5k	❩	= 6k
	100			100

∴ New ratio	❨	23k	:	33k	: 6k	❩	= 23 : 33 : 60
		10		10

Let the third number be x.

Then, first number = 120% of x =	120x	=	6x
	100		5

Second number = 150% of x =	150x	=	3x
	100		2

∴ Ratio of first two numbers =	❨	6x	:	3x	❩	= 12x : 15x = 4 : 5.
		5		2

Let the fourth proportional to 5, 8, 15 be x.

Then, 5 : 8 : 15 : x

⟹ 5x = (8 x 15)

x =	(8 x 15)	= 24.
	5

Suppose pipe A alone takes x hours to fill the tank.

Then, pipes B and C will take	x	and	x	hours respectively to fill the tank.
	2		4

∴	1	+	2	+	4	=	1
	x		x		x		5

⟹	7	=	1
	x		5

⟹ x = 35 hrs.

Let the cistern be filled by pipe A alone in x hours.

Then, pipe B will fill it in (x + 6) hours.

∴	1	+	1	=	1
	x		(x + 6)		4

⟹	x + 6 + x	=	1
	x(x + 6)		4

⟹ x² - 2x - 24 = 0

⟹ (x -6)(x + 4) = 0

⟹ x = 6.     [neglecting the negative value of x]

Part filled in 2 hours =	2	=	1
	6		3

Remaining part =	❨	1 -	1	❩	=	2	.
			3			3

∴ (A + B)'s 7 hour's work =	2
	3

(A + B)'s 1 hour's work =	2
	21

∴ C's 1 hour's work = { (A + B + C)'s 1 hour's work } - { (A + B)'s 1 hour's work }

=	❨	1	-	2	❩	=	1
		6		21			14

∴ C alone can fill the tank in 14 hours.

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]

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.