# Alligation

The allegation is an old and practical method of solving arithmetic problems related to mixtures of ingredients. There are two types of allegation: allegation medial, used to find the quantity of a mixture given the quantities of its ingredients, and allegation alternate, used to find the amount of each ingredient needed to make a mixture of a given quantity. Allegation medial is merely a matter of finding a weighted mean. Allegation alternate is more complicated and involves organizing the ingredients into high and low pairs which are then traded off.

## What is a Mixture?

mixture contains two or more ingredients of certain of certain quantity mixed together to get the desired quantity. The quantity can be expressed as ratios or percentage.

When two varieties of rice are mixed to form a new variety of rice then it is called as a mixture.

What is an Alligation?

If two ingredients A and B of price x and y respectively are mixed and the price of the resultant mixture is M (mean price)then the ratio (R) in which ingredients are mixed is given by, the rule of the alligation

An alligation is the rule that enables us to find the ratio in which two or more ingredients at a given price or value must be mixed to produce a mixture at a given price (Value or Strength).

Formulas for Mixtures and alligation questions Example 1Mixtures and Alligation Questions Question: How much of salt at 42 paise/kg must be mixed with 25kg of salt at 24 paise/kg, so that a profit of 25% is obtained by selling the mixture at 40 paise /kg? Solution: Th question is based on a combination of two different varieties of salt to obtain a mixture at the given price. In this question, we are suppose to find the quantity of salt which costs 42 paise that is mixed. So, we need to find the ratios of the quantity to find the quantity of salt that is mixed. Step 1: First, we need to find the price of the mixed salt, that is Profit = 25% Selling Price = 40 paise/kg Cost of the Mixture = 32 paise Step 2:

Use the pictorial representation method to find the ratios. Step 3: The ratio = 4 : 5 Quantity(dearer) : Quantity (Cheaper) = 4 : 5

Quantity(dearer) : 25 = 4 : 5

Quantity(dearer) x 5 = 4 x 25

Quantity(dearer) = 100/5

Quantity(dearer) = 20kg

Therefore, The 20kg of dearer quantity is supposed to be mixed with 25kg of cheaper quantity to make a new mixture of salt.