Alligation and
Mixtures Important formulas:
1. Alligation: 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 a desired
price.
Or, Alligation is an old and practical method of
solving arithmetic problems related to mixtures of ingredients.
2. Mean Price: The cost price of a unit quantity of the mixture is called the mean
price.
3. Rule of
Alligation: If two ingredients are mixed, i.e. if ‘c’ kg of cheaper thing
mixed with ‘d’ kg of dearer thing and we get ‘m’ kg of mixture, then
The Preparation for the solution is like this.
Therefore, (Cheaper quantity): (Dearer quantity) = (d  m): (m  c).
4. Suppose a container contains ‘x’ units of liquid from which ‘y’
units are taken out and replaced by water. After ‘n’ operations the quantity of
pure liquid=[x {1 (y/x)}^{n}] units.
Solved
Examples:
Example:
01
How much water
must be added to 60 litres of milk at 1½ liter for Rs. 20, so as to have a
mixture worth Rs.10⅔ a liter?
Solution:
C.P. of 1 liter of milk = Rs. (20 × 2/3) = Rs. 40/3
Therefore, Ratio of water and milk = (8/3): (32/3) = 1:
4
Hence, Quantity of water to be added
to 60 litres of milk = [(1/4) × 60] Liters
= 15 Liters.
Example: 02
How many kgs of wheat
costing Rs. 8 per kg must be mixed with 86 kg of rice costing Rs. 6.40 per kg
so that 20% gain may be obtained by selling the mixture at Rs. 7.20 per kg ?
Solution:
S.P. of 1 kg mixture = Rs. 7.20, Gain = 20%.
Therefore, C.P. of 1 kg mixture = Rs. [(100/120) × 7.20]
= Rs. 6.
By the rule of alligation, we have:
Wheat of 1st kind: Wheat of 2nd kind = 60: 200 = 3:
10.
Let ‘x’ kg of wheat of 1st kind be mixed with 36 kg of
wheat of 2nd kind.
Then, 3: 10 = x: 36 or 10x = 3 × 36 or x = 10.8 kg.
SOLVED EXAMPLES
PART 1
Question
No. 01
The milk and water in two
vessels A and B are in the ratio 4: 3 and 2: 3 respectively. In what ratio, the
liquids in both the vessels be mixed to obtain a new mixture in vessel C
containing half milk and half water?
Solution:
Let the C.P. of milk be Rs. 1 per litre
Milk in 1 litre mixture of A = 4/7 litre; Milk in 1
litre mixture of B = 2/5 litre;
Milk in 1 litre mixture of C = ½ litre
C.P. of 1 litre mixture in A = Rs. 4/7
C.P. of
1 litre mixture in B = Rs. 2/5
Mean price = Rs. 1/2
By the rule of alligation, we have:

Allegation and Mixture Solution01 
Therefore, required ratio = (1/10): (1/14) = 7: 5
Question
No. 02
In what ratio must water be
mixed with milk to gain 20 % by selling the mixture at cost price?
Solution:
Let C.P. of milk be Rs. 1 per litre.
Then, S.P. of 1 litre of mixture = Rs. 1.
Gain obtained = 20%.
Therefore, C.P. of 1 litre of mixture = Rs. [(100/120)
× 1] = 5/6
By the rule of alligation, we have:

Alligation and Mixture Solution02 
Therefore, Ratio of water and milk = (1/6): (5/6) = 1:
5.
Question
No. 03
In what ratio
must rice at Rs. 9.30 per kg be mixed with rice at Rs. 10.80 per kg so that the
mixture be worth Rs. 10 per kg?
Solution: By the rule of alligation, we have:

Alligation and Mixture Solution03 
(Cheaper quantity): (Dearer quantity) =
(d  m): (m  c).
=> (1080  1000): (1000  930) => 80: 70 =>8: 7
Therefore, required ratio =8: 7.
Question
No. 04
A vessel is filled with liquid, 3 parts
of which are water and 5 parts acid. How much of the mixture must be drawn off
and replaced with water so that the mixture may be half water and half acid?
Solution:
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/8) + x] litres
Quantity of acid in new mixture = [5 
(5x/8)] litres
Therefore, [3  (3x/8) + x] = [5 
(5x/8)]
=> 5x + 24 = 40  5x => 10x = 16
=> x = 8/5
So, parts of mixture replaced = (8/5) ×
(1/8) = 1/5.
Question
No. 05
Coffee
wroth 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:
Solution:
Since 1st and 2nd variety is mixed in
equal proportions.
So, their average price = Rs. [126 +
135)/2] = Rs. 130.50
So, the mixture is formed by mixing two verities,
one at Rs. 135.50 per kg and the other at say, Rs. x per kg in the ratio of 2 :
2, i.e. the price of 3rd kind is Rs. x.
By the rule of alligation, we have:

Al ligation and Mixture Solution05 
Therefore, (x  153) = 22.50 => x =
175.50
Hence, the price of 3rd variety per kg is
Rs. 175.50
Question
No. 06
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?
Solution:
By the rule of alligation, we have:

Alligation and Mixture Solution06 
Therefore, the required rate = 3.50: 1.50
= 7: 3
Question
No. 07
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:
Solution:
By the rule of alligation, we have:

Alligation and Mixture Solution07 
So, the ratio of 1st and 2nd parts = 4: 6
= 2: 3
Therefore, the quantity of 2nd kind =
(3/5) × 1000 = 600 kg.
Question
No. 08
A dishonest 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:
Solution:
Let, C.P of 1 litre milk be Rs. 1
Then, S.P of 1 litre of mixture = Rs. 1,
Gain = 25%
Therefore, C.P of 1 litre mixture = Rs.
(100/125) × 1 = 4/5
By the rule of alligation, we have:

Alligation and Mixture Solution08 
Therefore, ratio of milk to water = (4/5):
(1/5)
Hence, percentage of water in the mixture
= [(1/5) × 100] % = 20%
Question
No. 09
A jar full of water contains 40% sugar
dissolved in it. A part of the water is replaced by another containing 19% of
sugar and now the percentage of sugar dissolved was found to be 26%. The
quantity of sugar replaced is:
Solution:
By the rule of alligation, we have:

Alligation and Mixture Solution09 
So, the ratio of 1st and 2nd quantities =
7: 14 = 1: 2
Therefore, required quantity replaced =
2/3
Question
No. 10
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:
Solution:
Let the price of the mixed variety be Rs.
x per kg.
By the rule of alligation, we have:

Alligation and Mixture Solution10 
Therefore, (20  x)/ (x  15) = 2/3
=> (60  3x) = (2x  30)
=> 5x = 90
=> x = 18
Therefore, the price of the mixed variety
is Rs. 18 per kg.
PART 2
Question No. 01
A cup
contains milk and water in the ratio 3:1. How much mixture should be taken out
and water added to make the ratio 1:1?
(A) 1/3 part
of mixture
(B) 1/2 part
of mixture
(C) 1/4 part
of mixture
(D) 1/5 part
of mixture
Answer:
Option A
Explanation:
Let y part of mixture is replaced by water
Therefore,
(4x  y)/4x = 2x/3x
=> 12x²  3xy = 8x²
=> y = 4x/3
I.e. 1/3
part of mixture.
Question No. 02
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 liters of milk such that the ratio of water to milk
is 3:5?
(A) 4
liters, 8 liters
(B) 6
liters, 6 liters
(C) 5
liters, 7 liters
(D) 7
liters, 5 liters
Answer:
Option B
Explanation:
Let the cost
of 1 liter milk be Rs. 1
Milk in 1
liter mix in 1st can = 3/4 liter, C.P of 1 liter mix in 1st can Rs. 3/4
Milk in 1
liter mix in 2nd can = 1/2 liter, C.P of 1 liter mix in 2nd can Rs. 1/2
Milk in 1
liter of final mix = 5/8 liter, Mean price = Rs. 5/8
By the rule
of alligation, we have:
∴ Ratio
of two mixtures = (1/8) : (1/8) = 1 : 1
So, quantity
of mixture taken from each can = (½ × 12) = 6 liters
Question No. 03
In 2 gallons
mixture of spirit and water, the percentage of water is 12. In another 3
gallons mixture of spirit and water, the percentage of water is 5. These two
mixtures are poured in a third pot and ½ gallon more water is added to it. Find
the percentage of water in this new mixture.
(A) 17 ^{1}/_{11} %
(B) 17 ^{3}/_{11} %
(C) 16 ^{2}/_{11} %
(D) 16 ^{13}/_{27} %
Answer:
Option C
Explanation:
W_{1 }=
12% of 2 = 24/100 = 0.24
∴ W_{2}
= 5% of 3 = 0.15
W_{3} = 0.5
∴ W_{1}
+ W_{2} + W_{3} = 0.24 + 0.15 + 0.5 = 0.89
∴ New
W% = (0.89/5.5) × 100 = 16 ^{2}/_{11}
%
Question No. 04
How many kg
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.
Answer:
Option D
Explanation:
S.P of 1 kg
of mixture = Rs. 9.24, Gain 10%
∴ C.P
of 1 kg of mixture = Rs. [(100/110) × 9.24] = Rs. 8.40
By the rule
of alligation, we have:
∴ Ratio
of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3
Let, x kg of sugar of 1st kind be mixed with
27 kg of 2nd kind
Then, 7 : 3
= x : 27
=> x = (7 × 27)/3 = 63 kg.
Question No. 05
Three
identical vessels contain the mixture of spirit and water. The ratio of spirit
and water in each glass is 2:3, 3:4 and 4:5 respectively. The mixtures of all
the three vessels are poured into a big pot. The ratio of spirit and water in
the new mixture is
(A) 401/544
(B) 27/37
(C) 19/37
(D) 13/37
Answer:
Option A
Explanation:
Net spirit =
(2/5) + (3/7) + (4/9) = 401/315
∴ Net
water = 3  (401/315) = 544/325
Hence
required ratio = 401/544
Question No. 06
A container
contains 40 liters of milk. From this container 4 liters of milk was taken out
and replaced with water. This process was repeated further 2 times. How much
milk is now contained by the container?
(A) 26.34
liters
(B) 27.36
liters
(C) 28
liters
(D) 29.16
liters
Answer:
Option D
Explanation:
Amount of
milk left after 3 operations = [40 × {1  (4/40)}^{3}] liters
= [40 × (9/10) × (9/10) × (9/10)] = 29.16 liters
Question No. 07
In what
ratio must water be mixed with milk to gain 16⅔ % on selling the mixture at
cost price?
(A) 1 : 6
(B) 6 : 1
(C) 2 : 3
(D) 4 : 3
Answer:
Option A
Explanation:
Let C.P of 1
liter milk be Rs.1
S.P of 1
liter of mixture = Rs. 1, gain = (50/3) %
∴ C.P
of 1 liter of mixture = [100 × (3/350) × 1] = 6/7
By the rule
of alligation, we have:
∴ Ratio
of water and milk = (1/7) : (6/7) = 1 : 6
Question No. 08
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
Answer:
Option B
Explanation:
By the rule
of alligation, we have:
∴ Required
ratio = 60 : 90 = 2 : 3
Question No. 09
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
Answer:
Option A
Explanation:
S.P of 1 kg
of mixture = 68.20, Gain = 10%
C.P of 1 kg
of the mixture = Rs [(100/110) × 68.20] = Rs. 62
By the rule
of alligation, we have:
∴ Required
ratio = 3 : 2
Question No. 10
A can
contains a mixture of two liquids ‘A’ and ‘B’ in the ratio 7 : 5. When 9 liters
of mixture are drawn off and the can is filled with ‘B’, the ratio of ‘A’ and
‘B’ becomes 7 : 9. how many liters of liquid ‘A’ was contained by the can
initially?
(A) 10
(B) 20
(C) 21
(D) 25
Answer:
Option C
Explanation:
Suppose the
can initially contains 7x and 5x of mixtures A and B respectively
Quantity of
A in mixture left = [7x  (^{7}/_{12}
× 9)] liters = [7x  (21/4)] liters
Quantity of
B in the mixture left = [5x  (^{5}/_{12}
× 9)] liters = [5x  (15/4)] liters
∴ [7x  (21/4)]/ [5x  (15/4)] = 7/9
=> (28x  21)/(20x + 21) = 7/9
=> 252x  189 = 140x + 147
=> 112x = 336
=> x = 3
So, the can
contained 21 liters of A.
Question No. 11
8 liters 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
liters
(B) 24
liters
(C) 32
liters
(D) 42
liters
Answer:
Option B
Explanation:
Let the
quantity of the wine in the cask originally be x liters
Then,
quantity of wine left in the cask after 4 operations = [x {1  (8/x)}^{4}]
liters
[x {1  (8/x)}^{4}]/x =
16/81
=> {1 
(8/x)}^{4} = (2/3)^{4}
=> [(x  8)/x] = 2/3
=> 3x  24 = 2x
=> x = 24