# Alligations and Mixture Formulas

## Formulas For Alligation And Mixture

### Basics of Mixtures and Alligations:

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 to form the mixutre.

The most fundamental point to remember while solving mixtures and alligations is that alligation is a way to find the mean value of mixture when the ratio and amount of the ingredients mixed are different and also to find the proportion in which the elements are mixed.

The Alligation and Mixture Formulas can be applied to any topic like mixtures, profit and loss, simple interest, time and distance, percentage, etc.

## What is an Alligation?

When two ingredients X and Y of price p and q respectively are mixed together,  such that 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.

(Cheaper quantity) : (Dearer quantity) = (d – m) : (m – c)

## What is a Mixture?

In a mixture,  two or more ingredients are mixed together to get a desired quantity. The quantity can be expressed as ratio or percentage.

For example: When two varieties of sugar are mixed to form a new variety of sugar then it is called as a mixture.

## What is a Mean Price?

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

## Alligation and Mixture Formulas 1

When two commodities are mixed then ,

 Quantity of cheaper = C.P. of dearer(d) – Mean Price(m) Quantity of dearer Mean price(m) – C.P. of cheaper(c)

## Alligation and Mixture Formulas 2

Consider a container contain x unit of liquid A from which y units are taken out and replaced by water. This operation is repeated n number of times, then the quantity of pure liquid will be given by the formula:

$Quantity = \left[x – (1 – \left(\frac{Y}{X}\right)^n \right] units$

Therefore,  (Cheaper Commodity) : (Dearer Commodity) =  (d – m)  :  (m-c)

## Alligation and Mixture Formulas 3

Calculate quantity of pure Liquid after ‘n’ successive operations,

If a container contains ‘x’ units of pure liquid , and we replace the liquid with ‘y’  units of water ,

Then after ‘n’ successive operations, the units of pure liquid left is ,

 After n operations, the quantity of pure liquid = x 1 — y n units. x