## Friday, 21 October 2016

### Problems on Alligations and Mixtures

1. Allegation:
It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.
2. Mean Price:
The cost of a unit quantity of the mixture is called the mean price.
3. Rule of Allegation:
If two ingredients are mixed, then
 Quantity of cheaper = C.P. of dearer - Mean Price Quantity of dearer Mean price - C.P. of cheaper
We present as under:
 C.P. of a unit quantityof cheaperC.P. of a unit quantityof dearer (c) Mean Price(m) (d) (d - m) (m - c)
(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c).
4. Suppose a container contains x of liquid from which y units are taken out and replaced by water.
 After n operations, the quantity of pure liquid = x 1 - y n units. x
1.
A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
A.
 1 3
B.
 1 4
C.
 1 5
D.
 1 7
Explanation:
Suppose the vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.
 Quantity of water in new mixture = 3 - 3x + x litres 8
 Quantity of syrup in new mixture = 5 - 5x litres 8
 3 - 3x + x = 5 - 5x 8 8
5x + 24 = 40 - 5x
10x = 16
 x = 8 . 5
 So, part of the mixture replaced = 8 x 1 = 1 . 5 8 5
2.
Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:
 A. Rs. 169.50 B. Rs. 170 C. Rs. 175.50 D. Rs. 180
Explanation:
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 kindCost of 1 kg tea of 2nd kind Rs. 130.50 Mean PriceRs. 153 Rs. x (x - 153) 22.50
 x - 153 = 1 22.50
x - 153 = 22.50
x = 175.50
3.
A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?
 A. 10 B. 20 C. 21 D. 25
Explanation:
Suppose the can initially contains 7x and 5x of mixtures A and B respectively.
 Quantity of A in mixture left = 7x - 7 x 9 litres = 7x - 21 litres. 12 4
 Quantity of B in mixture left = 5x - 5 x 9 litres = 5x - 15 litres. 12 4
 7x - 21 4
=7
 5x - 15 + 9 4
9
 28x - 21 = 7 20x + 21 9
252x - 189 = 140x + 147
112x = 336
x = 3.
So, the can contained 21 litres of A.
4.
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?
 A. 4 litres, 8 litres B. 6 litres, 6 litres C. 5 litres, 7 litres D. 7 litres, 5 litres
Explanation:
Let the cost of 1 litre milk be Re. 1
 Milk in 1 litre mix. in 1st can = 3 litre, C.P. of 1 litre mix. in 1st can Re. 3 4 4
 Milk in 1 litre mix. in 2nd can = 1 litre, C.P. of 1 litre mix. in 2nd can Re. 1 2 2
 Milk in 1 litre of final mix. = 5 litre, Mean price = Re. 5 8 8
By the rule of alligation, we have:
C.P. of 1 litre mixture in 1st can   C.P. of 1 litre mixture in 2nd can
 3 4
Mean Price
 5 8
 1 2
 1 8
 1 8
 Ratio of two mixtures = 1 : 1 = 1 : 1. 8 8
 So, quantity of mixture taken from each can = 1 x 12 = 6 litres. 2
5.
In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs. 16.50 kg?
 A. 3 : 7 B. 5 : 7 C. 7 : 3 D. 7 : 5
Explanation:
By the rule of alligation:
 Cost of 1 kg pulses of 1st kindCost of 1 kg pulses of 2nd kind Rs. 15 Mean PriceRs. 16.50 Rs. 20 3.50 1.50
Required rate = 3.50 : 1.50 = 7 : 3.
6.
A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is:
 A. 4% B. 6% C. 20% D. 25%
Explanation:
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 milkC.P. of 1 litre of water
Re. 1Mean 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
7.
How many kilogram of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing Rs. 7 per kg so that there may be a gain of 10% by selling the mixture at Rs. 9.24 per kg?
 A. 36 kg B. 42 kg C. 54 kg D. 63 kg
Explanation:
S.P. of 1 kg of mixture = Rs. 9.24, Gain 10%.
 C.P. of 1 kg of mixture = Rs. 100 x 9.24 = Rs. 8.40 110
By the rule of allilation, we have:
 C.P. of 1 kg sugar of 1st kindCost of 1 kg sugar of 2nd kind Rs. 9 Mean PriceRs. 8.40 Rs. 7 1.40 0.60
Ratio of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3.
Let x kg of sugar of 1st be mixed with 27 kg of 2nd kind.
Then, 7 : 3 = x : 27
 x = 7 x 27 = 63 kg. 3
8.
A container contains 40 litres of milk. From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?
 A. 26.34 litres B. 27.36 litres C. 28 litres D. 29.16 litres
Explanation:
 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
9.
A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is:
A.
 1 3
B.
 2 3
C.
 2 5
D.
 3 5
Explanation:
By the rule of alligation, we have:
 Strength of first jarStrength of 2nd jar 40% MeanStrength26% 19% 7 14
So, ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2
 Required quantity replaced = 2 3
10.
In what ratio must water be mixed with milk to gain 16% on selling the mixture at cost price?
 A. 1 : 6 B. 6 : 1 C. 2 : 3 D. 4 : 3
Explanation:
Let C.P. of 1 litre milk be Re. 1.
 S.P. of 1 litre of mixture = Re.1, Gain = 50 %. 3
 C.P. of 1 litre of mixture = 100 x 3 x 1 = 6 350 7
By the rule of alligation, we have:
C.P. of 1 litre of waterC.P. of 1 litre of milk
0Mean Price
 Re. 6 7
Re. 1
 1 7
 6 7
 Ratio of water and milk = 1 : 6 = 1 : 6. 7 7
1.
Find the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a mixture worth Rs. 6.30 a kg.
 A. 1 : 3 B. 2 : 3 C. 3 : 4 D. 4 : 5
Explanation:
By the rule of alligation:
 Cost of 1 kg of 1st kindCost of 1 kg of 2nd kind 720 p Mean Price630 p 570 p 60 90
Required ratio = 60 : 90 = 2 : 3.
12.
In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that by selling the mixture at Rs. 68.20 a kg he may gain 10%?
 A. 3 : 2 B. 3 : 4 C. 3 : 5 D. 4 : 5
Explanation:
S.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10%.
 C.P. of 1 kg of the mixture = Rs. 100 x 68.20 = Rs. 62. 110
By the rule of alligation, we have:
 Cost of 1 kg tea of 1st kind.Cost of 1 kg tea of 2nd kind. Rs. 60 Mean PriceRs. 62 Rs. 65 3 2
Required ratio = 3 : 2.
13.
The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is:
 A. Rs. 18 B. Rs. 18.50 C. Rs. 19 D. Rs. 19.50
Explanation:
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 riceCost of 1 kg of Type 2 rice Rs. 15 Mean PriceRs. x Rs. 20 (20 - x) (x - 15)
 (20 - x) = 2 (x - 15) 3
60 - 3x = 2x - 30
5x = 90
x = 18.
14.
8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of water is 16 : 65. How much wine did the cask hold originally?
 A. 18 litres B. 24 litres C. 32 litres D. 42 litres
Explanation:
Let the quantity of the wine in the cask originally be x litres.
 Then, quantity of wine left in cask after 4 operations = x 1 - 8 4 litres. x
 x(1 - (8/x))4 = 16 x 81
 1 - 8 4 = 2 4 x 3
 x - 8 = 2 x 3
3x - 24 = 2x
x = 24.
15.
A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is:
 A. 400 kg B. 560 kg C. 600 kg D. 640 kg