Ratio and Proportion Tips and Tricks for Government Exams

Introduction

• Ratio is a comparison of two quantities by division. Ratio represents the relation that one quantity bears to the other. It is represented as a:b. In any ratio a:b, a is called Antecedent and B is called Consequent. It is an abstract (without units) quantity.
  A ratio remains unaltered if its numerator and denominator are multiplied or divided by the same number, e.g. 4:3 is the same as the (4 x 10) : (3 x 10) ie 40:30.
• Proportion is a statement that two ratios are similar. When two ratios are equal, they make a proportion, i.e. if a/b = c/d, then a, b, c and d are in proportion. This is represented as a:b :: c:d. When a, b, c and d are in proportion, then a and d are called the Extremes and b and c are called the Means, also Product of the Means = Product of the Extremes i.e. bc = ad.
  

Theory

Ratio is a quantity which represents the relationship between two similar quantities. It expresses a magnitude by which quantity is multiple of another one. Ratio is represented as 2:3 or 2/3.
Here, numerator i.e. 2 is known as "ANTECEDENT" and denominator i.e. 3 is known as "CONSEQUENT".
If antecedent is more than the consequent, then it is known as improper ratio and if less ,then it is known as proper ratio.

Different Types of Ratios

• Duplicate Ratio: a² : b² is called duplicate ratio of a : b.
• Triplicate Ratio: a³ : b³ is called triplicate ratio of a : b.
• Sub – Duplicate Ratio: √a :√b is called sub-duplicate ratio of a : b.
• Sub – Triplicate Ratio:∛a : ∛b is called sub-triplicate ratio of a : b Compound Ratio: ab : cd is the compound ratio of a : c and b : d. It is the ratio of the product of the antecedents to that of the consequents of two or more given ratios. Inverse Ratio: 1/a : 1/b is the inverse ratio of a : b.
• Componendo and Dividendo: If a/b = c/d, then (a + b)/(a – b) = (c + d)/(c – d)

Different Types of Proportion

Continued Proportion: If these quantities a, b and c are such that a:b :: b:c, then b² = ac and a, b and c are in continued proportion. Also the quantity c is called the third proportion of a and b.
Fourth Proportion: If four quantities a, b, c and x are such that a:b :: c:x, then ax = bc and x is called the fourth proportion of a, b and c.
Mean or second Proportion: If three quantities a, b and x are such that a:x :: x:b, then x² = ab and x is called the mean of a and b. Also, if a:b = c:d, then the following properties holds good.

• b:a = d:c (Invertendo) 
• a:c = b:d (Alternendo) 
• (a + b) : b = (c + d) : d (Componendo) 
• (a – b) : b = (c – d) : d (Divendendo) 
• (a + b)/(a – b) = (c + d)/(c – d) (Componendo – Divendendo)

Solved Examples