# Mixtures and Alligation

### Basics of Fractions OR Chain Rule

When a fraction has its numerator greater than the denominator, its value is greater than one. It is called greater fraction. Whenever a number (say x) is multiplied by a greater fraction, it gives a value greater than itself.

When a fraction has its numerator less than the denominator, its value is less than one. It is called less fraction. Whenever a number (say x) is multiplied by a less fraction, it gives a value less than itself.

### Mixtures & Alligation

In such type of questions, you deal with problems related to mixing of different compounds or quantities. The concept of alligation and weighted average are the same.

When two or more quantities are mixed together in different ratios to form a mixture, then ratio of the quantities of the two constituents is given by the following formula: Always identify the ingredients as cheaper and dearer to apply the alligation rule.

In the alligation rule, the variables c, d and m may be expressed in terms of percentages (e.g. A 20% mixture of salt in water), fractions (e.g. two-fifth of the solution contains salt) or proportions (e.g. A solution of milk and water is such that milk : Water = 2 : 3).

### Removal and Replacement

If a vessel contains “x” litres of milk and if “y” litres be withdrawn and replaced by water, then if “y” litres of the mixture be withdrawn and replaced by water, and the operation repeated ‘n’ times in all, then: 