# Mixtures and Alligations Concept

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Mixtures and alligations is a specific type of ratio and proportion concept. This Mixture and alligation concept consists of a limited number of concepts and learning with different tips and tricks can help in achieving success in solving problems asked in various competitive exams.

## What is Alligation?

Alligations means linking. It is a rule to find the ratio in which two or more ingredients at their respective prices be mixed to give a mixture of the desired price.

### Mean Price in Mixtures and Alligations Concept

The cost price of a unit quantity of the mixture is termed the mean price.

### Alligation Formula

Let us consider two ìngrediants A and B of prices x and y respectively are mixed to forms a resultant mixture of price M(mean price). The ratio(R) in which ingredients are mixed is given by the rule of alligation.
\begin{align*}
R= \dfrac {y-M}{M-x} =\dfrac {M-y}{x-M}
\end{align*}

## Remove and Replacement in Mixtures and Alligations Concept

Suppose a container contains a solution and some part of the solution is removed and replaced by another ingredient. Now, again some amount of the first solution is removed and replaced by the second ingredient. This process is repeated n times then,

### Let us understand this removal and replacement concept with an example

A 40-liter mixture of milk and water contains milk and water in the ratio 3: 2. 20 liters of the mixture is drawn off and can be filled with pure milk and this operation is repeated once more. what is the ratio of milk and water in the resultant mixture after this operation?

#### Solution:

Here we are essentially replacing the water in the mixture with pure milk.so, gradually the number of milk increases and water decreases.
The quantity of water in the mixture originally = 8 liters.
The initial total solution in the mixture = 40litres
Number of times the process repeated =2
Quantity removed each time =20litres
Applying the removal and replacement formula
Final quantity of water is
\begin{align*} 16 \times \left[ 1-\dfrac { 20}{40}\right] ^{2} \\ \\ =4 \ litres
\end{align*}
then amount of milk in 40 liters solution is 36liters
Ratio of resultant quantity of milk and water in the mixture after the process is done twice is 36: 4 or 9: 1

## Mixtures and alligations Easy Questions

Q1. Let a certain quantity of water is mixed with milk priced at Rs 12 per litre. If the price of mixture is Rs 8 per litre. Calculate the ratio of water and milk in the new mixture.

Solution: From the given data,
Cost Price of 1 litre of milk (M) is Rs. 12
Cost Price of 1 litre of water (W) is Rs. 0
Mean Price of milk and water is (m) Rs. 8

From above formula of mistures and alligation,
Water : Milk = (CP of Milk – Mean Price):(Mean Price – CP of Water)
Water : Milk = (M-m):(m-W)
⇒(12-8):(8-0)
⇒4:8
⇒1:2
Water : Milk =1:2
Therefore, the ratio of water and milk in the new mixture is 1:2.