Mixtures and Alligations: Solved Examples

# Mixtures and Alligations: Solved Examples | CSAT Preparation - UPSC PDF Download

## Mixtures & Alligation

• As the dictionary meaning of Alligation (mixing), we will deal with problems related to mixing of different compounds or quantities. The concept of alligation and weighted average are the same.
• When two or more quantities are mixed together in different ratios to form a mixture, then ratio of the quantities of the two constituents is given by the following formulae:

• Gives us the ratio of quantities in which the two ingredients should be mixed to get the mixture.

Example.1: A sum of Rs 39 was divided among 45 boys and girls. Each girl gets 50 paise, whereas a boy gets one rupee. Find the number of boys and girls.

• Average amount of money received by each =
• Amount received by each girl = 50 paise = Rs
• Amount received by each boy = Re. 1

Number of girls = 45 – 33 = 12.

Important Funda

• Always identify the ingredients as cheaper & dearer to apply the alligation rule.
In the alligation rule, the variables c, d & m may be expressed in terms of percentages (e.g. A 20% mixture of salt in water), fractions (e.g. two-fifth of the solution contains salt) or proportions (e.g. A solution of milk and water is such that milk : Water = 2 : 3).
• The important point is to remember is that c & d may represent pure ingredients or mixtures.

## Mixing a pure component into a solution

Example.2: A jar contains a mixture of two liquids A and B in the ratio 4 : 1. When 10 litres of the liquid B is poured into the jar, the ratio becomes 2: 3. How many litres of liquid A were contained in the jar?

Method 1: (weighted average or equation method)

• Let the quantities of A & B in the original mixture be 4x and x litres.
• According to the question
12x = 2x + 20
⇒ 10x = 20
⇒ x = 2
• The quantity of A in the original mixture = 4x = 4 × 2 = 8 litres.

Method 2: (Alligation with composition of B)

• The average composition of B in the first mixture is 1/5.
• The average composition of B in the second mixture = 1
• The average composition of B in the resultant mixture = 3/5
• Hence applying the rule of Alligation we have
[1 – (3/5)]/[(3/5) – (1/5)] = (2/5)/(2/5) = 1
So, initial quantity of mixture in the jar = 10 litres.
And, quantity of A in the jar = (10 × 4)/5 = 8 litres.

Method 3: (Alligation with percentage of B)

• The percentage of B in 1st mixture = 20%
• The percentage of B in 2nd mixture = 100%
• The percentage of B in Final Mixture = 60%
• By rule of allegation we have
Volume 1st : Volume 2nd = (100% - 60%) : (60% - 20%)
V1 : V2 = 1 : 1
Volume of mixture 1st = 10 litres
Volume of A in mixture 1st = 80% of 10 litres = 8 litres. Answer

Question for Mixtures and Alligations: Solved Examples
Try yourself: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?

Question for Mixtures and Alligations: Solved Examples
Try yourself: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:

### Removal and Replacement

Example.3: Nine litres of solution are drawn from a cask containing water. It is replaced with a similar quantity of pure milk. This operation is done twice. The ratio of water to milk in the cask now is 16 : 9. How much does the cask hold?

• Let there be x litres in the cask
• After n operations:

(1 – 9/x)2 = 16/25
∴ x = 45 litres.

Example.4: There are two containers A and B of milk solution. The ratio of milk and water in container A is 3 : 1 and in container B, it is 4 : 1. How many liters of container B solution has to be added to 20. lts of container ‘A’ solution such that in the resulting solution; the ratio of milk to water should be 19 : 6?

• In container A, the part of milk =
• It is given 20 lts of container A is added. So, the quantity of container B should be 5 lts.

Example.5: There are two alloys A and B. Alloy A contains zinc, copper and silver, as 80% 15% and 5% respectively, whereas alloy B also contains the same metals with percentages 70%, 20%, 10% respectively. If these two alloys are mixed such that the resultant will contain 8% silver, what is the ratio of these three metals in the resultant alloy?

• Since the resultant alloy contains 8% silver, first we will find, in what ratio these two alloys A and B were mixed to form the resultant.
• Then the resultant zinc percentage is
• So, copper percentage = 100 – (74 + 8) = 18
∴ The ratio of these metals = 74 : 18 : 8 = 37 : 9 : 4.

Example.6: The cost of an apple is directly proportional to square of its weight in a fruit bazaar. Two friends A and B went there to purchase apples. A got exactly 5 apples per kg and each apple is of same weight. Where as B got exactly 4 apples per kg each weight is exactly same. If B paid Rs. 10 more than A per kg apples, what is the cost of an apple which weighs 1 kg?

• It is given cost ∝ (weight)2
⇒ c = k w2.
• A got 5 apples per kg and each apple is of same weight. ⇒ Each apple is 200 gm. = 1/5 kg.
The document Mixtures and Alligations: Solved Examples | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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## FAQs on Mixtures and Alligations: Solved Examples - CSAT Preparation - UPSC

 1. What is the concept of mixtures and alligation?
Ans. Mixtures and alligation is a mathematical concept used to solve problems related to the mixing of different components to obtain a desired concentration or strength.
 2. How is mixtures and alligation useful in real life?
Ans. Mixtures and alligation concepts are widely used in various fields such as pharmacy, chemistry, cooking, and manufacturing industries, where accurate measurements and proportions are crucial for desired outcomes.
 3. Can you explain the basic formula or equation used in mixtures and alligation?
Ans. The basic formula used in mixtures and alligation is: (Quantity of Component A) / (Quantity of Component B) = (Price of Component B - Price of Mixture) / (Price of Mixture - Price of Component A) This formula helps in determining the ratio of different components in a mixture.
 4. How can mixtures and alligation be applied to solve practical problems?
Ans. To solve practical problems using mixtures and alligation, one must first determine the ratios of the components in the given mixture or solution. Then, by using the given information about the components or solutions being mixed, the desired concentration or strength can be calculated.
 5. Can you provide an example of mixtures and alligation in a real-life scenario?
Ans. Sure! Let's consider a scenario where a pharmacist needs to prepare a 10% saline solution. They have a 5% saline solution and a 15% saline solution. By using mixtures and alligation, the pharmacist can calculate the ratio of these two solutions to obtain the desired 10% saline solution.

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