A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5?

Let the price of the mixed variety be Rs. x per kg.

By rule of alligation, we have:

Cost of 1 kg of Type 1 rice Cost of 1 kg of Type 2 rice
Rs. 15	Mean Price Rs. x	Rs. 20
(20 - x)		(x - 15)

∴	(20 - x)	=	2
	(x - 15)		3

⟹ 60 - 3x = 2x - 30

⟹ 5x = 90

⟹ x = 18.

By the rule of alligation, we have:

Profit on 1st part Profit on 2nd part
8%	Mean Profit 14%	18%
4		6

Ration of 1^st and 2^nd parts = 4 : 6 = 2 : 3

∴ Quantity of 2nd kind =	❨	3	x 1000	❩kg	= 600 kg.
		5

Let the slower pipe alone fill the tank in x minutes.

Then, faster pipe will fill it in	x	minutes.
	3

∴	1	+	3	=	1
	x		x		36

⟹	4	=	1
	x		36

⟹ x = 144 min.

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.

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.

By the rule of alligation:

Cost of 1 kg pulses of 1st kind Cost of 1 kg pulses of 2nd kind
Rs. 15	Mean Price Rs. 16.50	Rs. 20
3.50		1.50

∴ Required rate = 3.50 : 1.50 = 7 : 3.

Since first and second varieties are mixed in equal proportions.

So, their average price = Rs.	❨	126 + 135	❩	= Rs. 130.50
		2

So, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say, Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x.

By the rule of alligation, we have:

Cost of 1 kg of 1st kind Cost of 1 kg tea of 2nd kind
Rs. 130.50	Mean Price Rs. 153	Rs. x
(x - 153)		22.50

∴	x - 153	= 1
	22.50

⟹ x - 153 = 22.50

⟹ x = 175.50

Amount of milk left after 3 operations =	[	40	❨	1 -	4	❩	3	] litres
					40

=	❨	40 x	9	x	9	x	9	❩	= 29.16 litres.
			10		10		10

Let C.P. of 1 litre milk be Re. 1

Then, S.P. of 1 litre of mixture = Re. 1, Gain = 25%.

C.P. of 1 litre mixture = Re.	❨	100	x 1	❩	=	4
		125				5

By the rule of alligation, we have:

C.P. of 1 litre of milk C.P. of 1 litre of water
Re. 1	Mean Price Re. 4 5	0
4 5		1 5

∴ Ratio of milk to water =	4	:	1	= 4 : 1.
	5		5

Hence, percentage of water in the mixture =	❨	1	x 100	❩%	= 20%.
		5

By the rule of alligation:

Cost of 1 kg of 1st kind Cost of 1 kg of 2nd kind
720 p	Mean Price 630 p	570 p
60		90

∴ Required ratio = 60 : 90 = 2 : 3.