Ratio and Proportions-Alligation

The ratio of two quantities ‘a’ and ‘b’ in the same unit, is the fraction a/b and it is denoted as a: b.

The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Example: 4: 5 = 8: 10 = 12: 15 etc.

Proportion:

The equality of two ratios is called proportion.

If a : b = c : d, also write as a : b : : c : d . ‘a’ and ‘d’ are called extremes, while ‘b’ and ‘c’ are called mean terms.

Product of means = Product of extremes.

Thus, a : b : : c : d Û (b  × c) = (a × d).

Points to remember:

1. Duplicate ratio of (a : b) is (a² : b²)
2. Sub-duplicate ratio of (a : b) is   Öa : Öb
3. Triplicate ratio of (a : b) is a³ : b³
4. Sub-triplicate ratio of (a : b) is  a^1/3 : b^1/3
5. If a/b = c/d , then  (a + b)/(a-b) = (c +d)/(c-d) (Componendo-Dividendo)

Variation:

1. We say that x is directly proportional to y, if x = ky for some constant k and we write, x α y.
2. We say that x is inversely proportional to y, if xy = k for some constant k and we write, x α 1/y.

Chain Rule

1. Direct Proportion: Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.

Example-1: Cost is directly proportional to the number of articles.  (More articles, More Cost).

Example-2: Work done is directly proportional to the number of men working on it. (More Men, More work).

2. Indirect Proportion: Two quantities are said to be indirectly proportional, if on the increase of the one, the other decrease to the same extent and vice-versa.

Example-1: The time taken by a car in covering a certain distance is inversely proportional to the speed of the car.

Example-2: Time taken to finish a work is inversely proportional to the number of person working at it. (More persons, Less is the time taken to finish a job).

Remark: In solving question by chain rule, we compare every with the term to be found out.

Alligation and Mixture :

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.
2. Mean Price: The cost price a unit quantity of the mixture is called the mean price.
3. Rule of Alligation: If two ingredients are mixed, then 

Suppose a container contains x units of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid =