# Alligations or MixturesAptitude Questions and Answers

## 1. 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 a desired price.

A process or rule for the solution of problems concerning the compounding or mixing of ingredients differing in price or quality.

Merrian - Webster Medical Dictionary

## 2. Mean Price

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

## 3. Basic Formula

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

$$R = \left(\dfrac{M-y}{x-M}\right)$$

The above formula can be represented in diagram below, which in turn is more intuitive to grasp

Ingredient A                        Ingredient B

(Price x)                             (Price y)

\                       /
Mean

(Price M)

/                        \

(M – y)           :             (x – M)

Thus the required ratio is,

$$R = \left(\dfrac{M-y}{x-M}\right) = \left(\dfrac{y - M}{M - x}\right)$$

## 4. Replacement of Part of Solution Formula

Suppose a container contains a solution from which some quantity of solution is taken out and replaced with one of the ingredients. This process is repeated n times then,

$\text{Final Amount of ingredient that is not replaced}=$

$$\text{Initial Amount} \times \left(\dfrac{\text{Vol. after removal}}{\text{Vol. after replacing}}\right)^n$$

Above formula is not only true for absolute amounts but for ratios as well. So following formula is also valid:

$\text{Final ratio of ingredient not replaced to total}=$

$$\text{Initial ratio} * \left(\dfrac{\text{Vol. after removal}}{\text{Vol. after replacing}}\right)^n$$

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