# Mixture and Alligation

In Mixture & Alligation, we study mixtures of two or more than two quantities that have different selling process or cost prices. Imagine that you mix a cheap substance with an expensive substance. What should be the value of the resulting mixture? That is what we will study in the Alligation & Rule of Alligation.

A mixture, as the name suggests is mixing two or more things together and alligation enables us to find the ratio in which the ingredients/ things have been mixed and at what price they are sole to earn profit or face loss.

To solve mixture and alligation questions, one must know that alligation is used to find the mean value of a mixture when the ratio and amount of the ingredients mixed are different and also to find the proportion in which the elements are mixed.

To understand the formula of the Alligation and the rule of alligation, we have to understand the concept of weighted average. For example, let us say that we buy 50 packs of chips at the cost of 10 rupees each and 30 packets at the cost of 20 rupees each. What is the average cost? Will the average will be determined by the 20 rupees packs or the 10 rupees packs? This is when we come across the concept of weighted average.

The weighted average here will be: [50×10]+[30×20]/80

In alligation, we will use the same concept. Let us first define the few following terms:

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 the desired price.

Mean Price: The cost price of a unit quantity of the mixture is called the mean price. Now let us define the rule of alligation.

## Rule Of Alligation

Let us suppose that two ingredients of concentrations ‘a’ and ‘b’ respectively have been mixed in some proportion. Let ‘a’ be the cheaper component and ‘b’ be the dearer or the most costlier component. Then the rule of alligation states that:

[{Quantity of Cheaper substance}/{Quantity of dearer substance}] = [(C.P. of dearer substance) – (Mean price)/(Mean price) – (C.P. of cheaper substance)]

Let ‘c’ be the cost price or C.P. of a unit quantity of a cheaper substance, ‘m’ be the mean price, ‘d’ be the cost price of a unit quantity of the dearer substance, then we can write:

(Quantity of the Cheaper Substance) : (Quantity of the Dearer Substance) = (d – m) : (m – c).

Example : In what ratio must rice at Rs 9.30 per kg be mixed with rice sold at Rs. 10.80 per kg, so that the mixture be worth Rs. 10 per kg?

Answer: Using the rule of alligation, we have:

C.P. of 1 kg of rice (in paise) = 1080 paise = d

Also the C.P. of 1 kg rice of 2nd kind (in paise) = 930 paise = c

Also, mean price of the mixture (per kg in paise) or m = 1000 paise. So from the rule of alligation, we have:

(Quantity of Cheaper rice) : (Quantity of dearer rice) = (1080 – 1000)/(1000 – 930)

Therefore the required ratio = 80 : 70 or 8:7 and hence the answer is 8: 7

### Important Formulas for Mixture and Alligation

- The basic formula which is used to find the ratio in which the ingredients are mixed is

It is also called the **rule of alligation** and can also be represented as

## The Method Of the Repeated Dilutions

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/n)}^{n}] units.

Example : 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?

Answer: The container contains x = 40 litres of milk. The quantity of milk that is taken out and replaced by water = y = 4 litres.

Also, we have been given that number of times the process is repeated or n = 3 (=1 +2 times). Therefore from the method of repeated dilution, substituting the relevant values, we have:

Amount of milk left after three operations = [40{ 1 – (4/40)}^{3}] litres. Therefore we may write:

[40×(9/10)×(9/10)×(9/10)] = 29.16 litres. Therefore the answer is 29.16 litres.