Ratio and Proportion Introduction

Ratio
  1. A ratio is a number, so to find the ratio of two quantities, they must be expressed in the same units.
  2. Ratio compares two or more things of same kind.
  3. If the two quantities are a and b, then ratio of a and b represented as a : b or (a / b)
  4. Here a is called antecedent and b is called consequent.
  5. A ratio does not change if both of its terms are multiplied or divided by the same number. Thus $\cfrac{2}{3}=\cfrac{10}{15}$
  6. Comparison of Ratios : We say that (a : b) > (c : d) $\Rightarrow \cfrac{\text{a}}{\text{b}}>\cfrac{\text{c}}{\text{d}}$
Types of Ratios
  1. Duplicate ratio : The ratio of squares of two number is called the duplicate ratio of the two numbers.
    In General : Duplicate ratio of (a : b) is $(a^2:b^2)$
    Example : Duplicate ratio of 1 : 2 is 2 : 4
  2. Sub-duplicate ratio : The ratio of square root of two number is called the sub-duplicate ratio of the two numbers.
    In General : Sub-duplicate ratio of (a : b) is $(\sqrt{a}:\sqrt{b})$
    Example : Sub-duplicate ratio of 9 : 16 is 3 : 4
  3. Triplicate ratio : The ratio of cubes of two number is called the triplicate ratio of the two numbers.
    In General : Triplicate ratio of (a : b) is $(a^3:b^3)$
    Example : Triplicate ratio of 2 : 4 is 8 : 64
  4. Sub-triplicate ratio : The ratio of cube root of two number is called the sub-triplicate ratio of the two numbers.
    In General : Triplicate ratio of (a : b) is $(a^{\cfrac{1}{3}}:b^{\cfrac{1}{3}})$
    Example : Sub-triplicate ratio of 8 : 64 is 2 : 4
  5. Compound ratio : The ratio of the product of antecedent to that of the consequent of two or more given ratio is called the compound ratio.
    Example 1 : Compound ratio of A : B and C : D is AC : BD.
    Example 2 : The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
  6. Inverse ratio : When we replace the antecedent with consequent and consequent with antecedent then we get inverse ratio.
    Example : Inverse ratio of A : B is B : A

    Note :

    If $\cfrac{a}{b} =\cfrac{c}{d}$ then $\cfrac{a+b}{a+b}$ $=\cfrac{c+d}{c-d}$ [Componendo and Dividendo]

Proportion
  1. Proportion : The equality of two ratios is called proportion.
    If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
    Here 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, (b x c) = (a x d)
  2. Fourth Proportional :
    If a : b = c : d, then d is called the fourth proportional to a, b, c.
  3. Third Proportional :
    a : b = c : d, then c is called the third proportion to a and b.
  4. Mean Proportional :
    Mean proportional between a and b is $\sqrt{\text{ab}}$.
  5. Variations:
    We say that x is directly proportional to y, if x = ky for some constant k and we write, $\text{x}\ \alpha \ \text{y}$
    We say that x is inversely proportional to y, if xy = k for some constant k and we write, $\text{x} \ \alpha \ \cfrac{1}{\text{y}}$