Mixture and Allegation

Mixture and Alligation

MIXTURE

When two or more than two substances are mixed in any ratio to produce a product, then the product is known as mixture.

Mean Price

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

ALLIGATION

Alligation literally means 'linking'. It is a rule to find:

(i) the ratio in which two or more ingredients at their respective prices should be mixed to given a mixture at a given price.

(ii) the mean or average price of a mixture when the prices of two or more ingredient which may be mixed together and the proportion in which they are mixed are given.

Rule of Alligation

Let the cost price of a unit of cheaper article is Rs. c and that of a unit of costly article is Rs. d and the average (mean) price of mixture is Rs. m, then

\[\frac{\text{Quantity}\,\,\text{of}\,\,\text{cheaper}\,\,\text{article}}{\text{Quantity}\,\,\text{of}\,\,\text{costly}\,\,\text{article}}\]

\[\text{=}\frac{\text{Cost}\,\,\text{price}\,\,\text{of}\,\,\text{a}\,\,\text{unit}\,\,\text{of}\,\,\text{costly}\,\,\text{article-Mean price}}{\text{Mean}\,\,\text{price}-\text{Cost}\,\,\text{price}\,\,\text{of}\,\,\text{a}\,\,\text{unit}\,\,\text{of}\,\,\text{a}\,\,\text{cheaper}\,\,\text{article}}\]

The above relation is represented as

Hence, quantity of cheaper : Quantity of dearer

                        \[=(d-m):(m-c)=\frac{d-m}{m-c}\]

Alligation rule is also used to find the ratio in which two or more ingredients at their respective prices should be mixed to produce a mixture at a given price.

Quicker One

Ø  A mixture contains two liquids in the ratio a : b. If x L of b is added to the mixture and the ratio of two liquids becomes a : c, then quantity of liquid a and that of liquid b is given by \[\frac{bx}{c-b}\] and \[\frac{ax}{c-b},\] respectively.

Ø  A container initially contains x units of a liquid. If a unit of liquid is taken out and it is filled with a unit of water repeatedly upto n times, then the final quantity of the original liquid in the container is given as \[\left[ x{{\left( 1-\frac{a}{x} \right)}^{n}} \right]\] units.

Ø  A container has milk and water in the ratio a : b, a second container has milk and water in the raito c : d. If both the mixture are emptied into a third container, then the ratio of milk to water in third container is given by \[\left[ \frac{a}{a+b}+\frac{c}{c+d} \right]:\left[ \frac{b}{a+b}+\frac{d}{c+d} \right]\]