Ratio and Proportion I

Introduction

      If a and b are any two quantities of the same kind a and b are not equal to zero. then the quotient a/b is called the ratio of a and b. It is expressed as a : b.
(read as a is to b).

Definition of ratio

     If a and b are two quantities with the same units such that b is not equal to zero; then the quotient a/b is called the ratio between a and b. It is written as a:b.
Ratio does not have any unit.
The quantities a and b are called terms of the ratio.
a' is called antecedent (first term)
b' is called consequent (second term).

Commensurable and incommensurable quantities

     Two quantities are said to be commensurable if their ratio can be expressed as the ratio of two integers. Other wise it is incommensurable.

Comparison of ratios

     Ratio of greater inequality, less inequality and equality:
A ratio a:b is said to be of
(1). greater inequality, if a>b (2). less inequality, if a < b (3). equality, if a=b.

Composition of ratios

     Compounded ratio: If two or more ratios are multiplied term-wise, the ratio thus formed is called their compounded ratio.

     Duplicate ratio: The compounded ratio of two equal ratios is called their duplicate ratio.

     Triplicate ratio: The compounded ratio of three equal ratios is called their triplicate ratio.

     Sub- Duplicate ratio: A ratio x:y is the sub-duplicate ratio of a:b. If a:b is the duplicate ratio of x:y.

     Sub triplicate ratio: A ratio x:y is the sub triplicate ratio of a:b if a:b is the triplicate ratio of x:y.

Illustrative Examples

     1. Find the compounded ratio of
a) 5:14 and 7:15
b) a+b:a-b; (a²-b²):(a²+b²) and a²+b²:(a+b)².

     2. In the ratio 7:8, if the consequent is 40, what is the antecedent?

     3. Find a) reciprocal ratio of 3:5 b) duplicate ratio of 2:3 c) triplicate ratio of 3:4 d) sub duplicate ratio of 25:16 e) sub triplicate ratio of 27:8

Proportion

     Four quantities a, b, c and d are said to be in proportion, if a:b=c:d. a and d are called extremes of the proportion. b and c are called means of the proportion.

Properties of proportions

     The Properties of proportions are: Invertendo, Alternendo, Componendo, Dividendo, Componendo and Dividendo.

Summary

     1. In a:b, a is called antecedent and b is called consequent.

     2. Two or more quantities are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third and so on.
a, b, c are in continued proportion if a:b=b:c.