quantitative aptitude- mixture & allegation applicable to Common Aptitude Test (CAT) Bank Competitive Exam UPSC Competitive Exams SSC Competitive Exams Defence Competitive Exams L.I.C/ G. I.C Competitive Exams Railway Competitive Exam University Grants Commission (UGC) Career Aptitude Test (IT Companies) and etc.

1. MIXTURE &
ALLIGATION
Suppose we have 5 L of water and 2.5 L of pure milk. Let us
try to answer the following question

2. Q. 1. If they are mixed to give a 7.5 L solution then what is ratio
between the quantity of milk and the water in it?
Q 2. If we want to make a solution which has only 80% milk
than in what ratio should we mix the milk and the water?

3. RULE OF ALLEGATION
When we mix a cheaper (lesser cost price) sample and a dearer (more
cost price) sample, the cost price of the mixture lies between the cost
price of the two. This cost price is called the mean price. All these
quantities are related as in the following formula :-

4. It can also be represented as :
CP of unit quantity
of cheaper (a )
CP of unit quantity
of dearer (b)
Mean price
(x)
(b - x) ( x – a)
(cheaper quantity) : (Dearer quantity)
=(b – x) : (x-a)
Make yourself comfortable with this visualization. This is very helpful

5. Examples
• Anil buys two varieties of sugar costing Rs. 50 per kg
and Rs. 20 per kg. He mixes these two varieties in a
certain ratio that cost him Rs. 18 per kg. Find the
ratio of cheaper quantity to that of the dearer
quantity.
• Akas buys a certain varieties of rice and mixes it
with other varieties of rice costing Rs. 25 per kg in
the ratio of 5 : 4. If the mixture cost Rs. 20 per kg,
what is the cost price of the other varieties?

6. • A dealer buys 11 kg of wheat for Rs. 275 and mixes it with another
quality of wheat in the ratio of 4 : 5. If the price of resulting
mixture is Rs. 30 per kg. What is cost price of other the quality of
wheat?
• A trades mixes two varieties of apples, costing Rs. 24 per dozen
and Rs. 36 per dozen and sells them at a rate of Rs. 33 per dozen,
there by gaining 10% on the transaction. The ratio in which he
mixes these varieties is :
• The ratio in which 30% alcohol solution should be mixed with
50% solution in order to get a 42% solution is:
• 11 L of water is mixed with a certain quantity of milk such that the
mixture is costs Rs. 1 per litre. If the cost of pure milk is 110 paise
per litre, then the amount of milk in the mixture is:

7. • Sheetal removed a certain quantity from a solution of 70% milk and
replace it by pure water. The concentration of resulting mixture is now
60%. What fraction of milk did Sheetal removed?
• Rashmi has 40 cups. She sells some at a profit of 15% and the rest at a
loss of 5%. She gains 10% on the entire transaction. What is the number
of cups she sells at a profit of 15%?
• A man sells two cycles for Rs. 2100 and gains 5% in the transaction. He
incurred a loss of 3 % on one and the profit of 7% on the other. Find the
cost of each cycle.
• A property dealer purchases two flats for Rs. 60 lacks. He sells one at a
loss of 5% and the other at a gain of 10 %. He losses 10 % on the entire
transaction. What is the cost of each flat?

8. • A container contains 100 kg of milk. From this container 10 kg of milk
was taken out and replaced by water. This process was further repeated
three times. How much milk does the container have now?
• The ratio of number of girls to number of boys is 1 : 2. If the average
weight of the boy is 30 kg and the average weight of the both the boys
and girls be 25 kg. then the average weight of the girls is :
• Two verities of soda water with the different price are mixed in the ratio
of 2 : 3. The price of first soda water is Rs. 10 per litre while the price of
second soda water is Rs. 15 per litre, respectively. The average price of
the mixture per litre is :
• 5 kg. of superior quality of sugar is mixed with 25 kg of inferior quality
of sugar. The price of superior and the price of inferior quality of sugar is
Rs. 18 and Rs. 12 respectively. The average price per kg of mixture is :

9. • 16 litre of kerosene is mixed with 5 litre of petrol. The price of kerosene is
Rs. 12 per litre and the price of petrol is Rs. 33 per litre. The average
price of mixture per litre is :
• A milkman has two types of milk. In the first container the %age of milk
is 80% and the second container the %age of milk is 60%. If he mixes
28L of milk of the first container to the 32 litre of milk to the second
container, is %age of milk in the mixture is :
• Modern electronic shop sold the 30% hardware at the profit of 50% and
90% software at the profit of 10%. The average profit % of the Modern
electronic shop is, if he sells only these two kinds of things:
• A mixture of rice is sold at Rs. 3 per kg. this mixture is formed by mixing
the rice of Rs. 2.10 and Rs. 2.52 per kg. what is the ratio of price of
cheaper to the costlier quality in the mature if the profit of 25% is being
earned ?

10. • A milkman has 20 litre of milk. If he mixes 5 litre of water, which
is freely available, in 20 litres of pure milk. If the cost of pure milk
is Rs. 18 per litre, then the profit of the milkman, when he sells all
the mixture at a cost price, is :
• In what ratio should water and wine be mixed so that after selling
the mixture at the cost price a profit of 33.33% is made?
• In what ratio should freely available water be mixed with the wine
worth Rs. 60 per L so that after selling the mixture at Rs. 50 per
litre, the profit will 25%
• A mixture of water and milk contains 80% milk. In 50 L of such a
mixture, how many litres of water is required to increase the %age
of water to 50%?

11. • Sharabi chand purchased two different kind of alcohol. In the first
mixture the ratio of alcohol to water is 3 : 4 and in the second
mixture it is 5 : 6. If he mixes the two given mixtures and makes a
third mixture of 18 L in which the ratio of alcohol to water is 4 : 5,
the quantity of first mixture (whose ratio is 3 : 4) is required to
make the 18 L of the third kind of mixture is :
• Some amount of Rs. 6000 was lent out 10% and the rest amount @
20% per annum and thus in 4 years the total interest from both the
amount collected was Rs. 3400. What is the amount which was lent
out@ 10% per annum?
• From a tank of petrol which contains 200 L of petrol, the seller
replaces each time with kerosene when he sells 40 L of petrol (or
its mixture). Every time he sells out only 40 L of petrol (pure or
impure). After replacing the petrol with kerosene 4th time, the total
amount of kerosene in the mixture is :

12. • From a container of wine, a thief has stolen 15 L of wine
and replaces it with some quantity of water. He again
repeated the same process. Thus in three attempts the
ratio of wine and water became 343 : 169. The initial
amount of wine in the container was :
• A jar was full with honey. A person used to draw out
20% of the honey from the jar and replaced it with
sugar solution. He has repeated the same process 4
times and thus there was only 512 gm honey left in the
jar, the rest part of the jar was filled with the sugar
solution. The initial amount of honey in the jar was :