Ratio is comparison of two or more similar quantities. The comparison means that one quantity is what multiple or fraction of the other. It is the relation in which one quantity bears to another of the same kind.

The ratio of P to Q is usually written as P : Q, the quantities A and B are called the terms of the ratio. The first term is called the antecedent, and the second term the consequent.

To find what multiple or part, A is of B, we divide A by B; hence the ratio A : B may be measured by the fraction. Fractions and ratios are same; the only difference is that ratio is a unit less quantity while fraction is not.

1. A ratio is said to be a ratio of greater inequality if antecedent is greater than the consequent.
2. A ratio is said to be a ratio of lesser inequality if antecedent is less than the consequent.
3. A ratio is said to be a ratio of equality if antecedent is equal to the consequent.

Two ratios can be compared by taking equivalent fractions to a common denominator. Thus suppose we have to compare \(\frac{3}{4}\) and\(\frac{5}{6}\).

\(\frac{3}{4}\) = \(\frac{9}{{12}}\)

\(\frac{5}{6}\) = \(\frac{{10}}{{12}}\) , now denominators are equal, 

Hence \(\frac{{10}}{{12}}\) \(\left( {or\,\,\,\frac{5}{6}} \right)\) is bigger

The ratio of two fractions can be expressed as a ratio of two integers.  Thus a ratio of \(\frac{x}{y}:\frac{u}{v}\) can be written as \(xv:uy\), where \(x, y, u\) and \(v\) are positive integers.

If either, or both, of the terms of a ratio be a surd quantity, then no two integers can be found which will exactly measure their ratio. Thus the ratio \(\sqrt 5 :1\) cannot be exactly expressed by any two integers.

Ratios are compounded by multiplying together the fractions which denote them; or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. A relation and ratio among the quantities follow inverse relationship, for example if

\(2a = 3b = 4c = 5d\)

\( \Rightarrow a : b : c : d\) = \(\frac{1}{2}:\frac{1}{3}:\frac{1}{4}:\frac{1}{5}\) 

= 30 : 20 : 15 : 12

Similarly if \(a : b : c : d = 1 : 2 : 3 : 4\), then 

\(\frac{a}{1} = \frac{b}{2} = \frac{c}{3} = \frac{d}{4}\) or \(12a = 6b = 4c = 3d\)

Example: The three sides of a triangle are in the ratio of 3 : 4 : 6, find the ratio of the three corresponding altitudes.

Solution: Suppose the three sides are \(3k, 4k\) and \(6k\). If the heights are \(a, b\) and \(c\), then area of this triangle 

 =\(\frac{{(3k)a}}{2} = \frac{{(4k)b}}{2} = \frac{{(6k)c}}{2}\)

or \(3a = 4b = 6c\),

Hence \(a : b : c =\frac{1}{3}:\frac{1}{4}:\frac{1}{6}\)= 4 : 3 : 2