## Alligation

Alligation method is basically for two purpose:

- to find the
**mean value**of the mixtures when prices and proportions of both the ingredients are given - to find the
**proportion**of an ingredients of mixture when prices of both ingredients and mixture are given

### Rule of Alligation

Let two ingredients are mixed, out of that one is **dearer** and other is **cheaper** comparatively. Let Cost price (C.P.) of dearer one = d and Cost price of cheaper one = c

Study the following figure, if you remember this, you can easily solve the questions related to this chapter. Always remember that you will always consider cost price, not the selling price

Therefore, (d-m) : (m-c) = (Cheaper Quantity) : (Dearer Quantity) So, you can easily find the ratio of both the ingredients if prices of both the ingredients are given.

## Examples with Solution

**Example1:** In what proportion rice @Rs3/kg be mixed with rice @ Rs 3.50/kg, so that mixture worth Rs. 3.15/kg?

**Solution:** C.P. of cheaper rice = Rs 3/kg C.P. of dearer rice = Rs 3.50/kg Mean price i.e. price of mixture = Rs 3.15/kg

Applying the Alligation method,

Therefore, Proportion in which both will be added is 7:3

**Example2:** A certain quantity of milk is mixed with 20 litres of water and the worth becomes Rs 15/litre. If price of pure milk is Rs 20/litre, then how much milk is there in the mixture?

**Solution:** Mixture = MILK + WATER

** **C.P. of pure milk = Rs 20/ltr

C.P. of water = Rs 0 Mean price i.e. price of mixture = Rs 15/ltr

Applying the Alligation method,

Ratio of water to milk = 1:3 Therefore, quantity of milk in mixture = 3 times quantity of water i.e. 3 times 20 = 60 litres

**Example3:** In what ratio must water be added to milk to gain 20% by selling the mixture at cost price?

**Solution:** In this ques, we will assume the cost price of milk so that we can simplify the problem.

Let assumed price **(C.P.) of milk = Rs 10/litre**

Then, according to ques, S.P. of milk = Rs 10/litre Gain by selling mixture = 20% Therefore selling price of mixture = Rs 120

C.P. = (100/120) times 10 ⇒**C.P. of mixture = Rs(25)/3 **

By alligation method, Ratio of water and milk = 1: 3

So, in this way we can calculate these types of problems. One more thing, if in case **percentage is given instead of cost price**, then also you can do the problem in the same way.

Hope you understand the topic. Will soon update more examples of related topic.