RATIOS, PROPORTIONS AND MIXTURES GATE 2020 STUDY MATERIAL PDF CIVILENGGFORALL

Ratio
Proportion
Mixtures and Allegations
Exercises, Problems, Solutions and hints on the above topics

RATIO

Ratio of two terms ‘a’ and ‘b’ is denoted by a: b (read as ‘a is to b’) and is measured by a b . Numerator ‘a’ is called ‘antecedent’ and denominator ‘b’ is called ‘consequent’. While comparing two quantities in terms of ratio:

Two quantities must be of the same kind.
Units of measurement of the two quantities must be the same. e.g., if a = 200 and b = 50 paisa, then a: b = 200 : 50 = 4 : 1.
Ratio is a pure number, i.e., without any unit of measurement as it denotes how many times a given quantity is in comparison to another.
Ratio would stay unaltered even if both antecedent and the consequent are multiplied or divided by the same number.

Properties of Ratios

Invertendo: If a / b = c / d, then b / a = d / c i.e., inverse ratios of two equal ratios are equal.
Alternendo: If a / b = c / d, then a / c = b / d i.e., ratios of antecedents and consequents of two equal ratios are equal.
Componendo: If a / b = c / d then a + b / b = c + d / d i.e., adding 1 to both sides.
Dividendo: If a / b = c / d, then a – b / b = c – d / d i.e., subtracting 1 from both sides.
Componendo-Dividendo: If a + b = c + d, t hen a + b / a – b = c + d / c – d i.e., dividing results of componendo by dividend.

PROPORTION

When ratio of two terms is equal to the ratio of two other terms, then these four terms are said to be in proportion, i.e., if a : b = c : d, then a, b, c and d are in proportion. a, b, c and d are called first, second, third and fourth proportional respectively. The terms a and d are called extremes, while b and c are called means. If a : b = c : d, then ad = bc i.e., product of extremes is equal to product of means.

MIXTURES AND ALLEGATIONS

Simple Mixture : When two different ingredients are mixed together, it is called simple mixture.

Compound Mixture : When two or more simple mixtures are mixed together to form another mixture, it is called compound mixture.

Allegation Rule : Allegation rule states that “When different quantities of the same or different ingredients, of different costs (one cheap and other dear) are mixed together to produce a mixture of a mean cost, then ratio of their quantities is inversely proportional to the difference in their cost from the mean cost.”

Application of Allegation Rule:

To find mean value of a mixture when prices of two or more ingredients, which are mixed together and the proportion in which they are mixed are given.
To find proportion in which ingredients at given prices must be mixed to produce a mixture at a given price.