# Ratio and Proportion | Introduction & Concept

__RATIO__

The ratio of two quantities a and b is the fraction ^{a}⁄_{b} and is expressed as a:b. Here ‘a’ is the numerator and ‘b’ is the denominator. Since the ratio expresses the number of times one quantity contains the other, it is an abstract (without units) quantity.

A ratio remains unaltered if its numerator and denominator are multiplied or divided by the same number.

Eg:- 7:6 is the same as (7 x 10) : (6 x 10) ie. 70:60

28:24 is the same as [^{28}⁄_{4}]:[^{24}⁄_{4}] ie , 7:6

“A ratio is the relationship between any two numbers that indicates how many times the first number contains the second.”

- a:b is called a ratio of greater inequality, If a > b (Eg, 5:4, 7:6, 9:4,...)
- a:b is called a ratio of less inequality, If a < b (Eg, 4:9, 2:3,1:8,....)
- a:b is called a ratio of equality, If a = b (Eg, 4:4, 7:7, 2:2,....)

## Types of Ratios

**Duplicate Ratio**: The duplicate ratio of a:b is a^{2}:b^{2}**Triplicate Ratio**: The triplicate ratio of a:b is a^{3}:b^{3}**Sub-duplicate Ratio**: The sub-duplicate ratio of a:b is √a : √b- Sub-triplicate Ratio The sub-triplicate ratio of a:b is ∛a : ∛b
**Compound Ratio**: The compound ratio of a:c and b:d is ab:cd . It is the ratio of the product of the numerators to that of the denominators of 2 or more given ratios.**Inverse Ratio**: The inverse ratio of a:b is^{1}⁄_{a}:^{1}⁄_{b}

__PROPORTION__

- If two ratios are equal they are called in proportion. For example: if
^{a}⁄_{b}=^{c}⁄_{d}, then a, b, c and d are in proportion. - When a,b,c and d are in proportion, then a and d are called Extremes and b and c are called Means.

Also, Product of Extremes = Product of Means, i.e. bc =ab

## Types of Proportion

**Continued Proportion**: If the ratio between the first and the second is equal to the ratio between the second and the third. That ratio is called continued proportion. i.e. a, b and c are in continued proportion, if a : b = b : c , Then, b^{2}= ac**Fourth Proportion**If a,b,c,d be four numbers such that a : b = c : d, then d is called the fourth proportional of a, b and c. Then ad = bc.**Mean and Second Proportion**If three numbers a,b and c are such that a : c :: c : b, then c^{2}=ab , and c is called the mean proportional of a and b . Formulae of mean proportion is , in this case c = √ab

__VARIATION__

If two numbers a and b are related in such a way that as the number a changes it also brings changes the second number b, then the two numbers are in variation.

This can be explained by an example of simple equation a = nb where n is a constant. If we assume that the value of n as 6 then the equation becomes as a = 6b.

When a = 1, b = 1 × 6 = 6

When a = 2, b = 2 × 6 = 12

When a = 3, b = 3 × 6 = 18

## Types of Variation

**Direct Variation**: The number ‘a’ is in direct variation with ‘b’ if an increase in ‘a’ makes simultaneous increase in ‘b’ proportionally and vice versa. It can be expressed as a=kb, where k is called the constant of proportionality.

Eg:- Perimeter of circle C= 2πr where 2 and π are constants and C increases if increases, decreases if r decreases. So C is in direct variation with r. Inverse Variation: The number ‘a’ is in inverse variation with ‘b’ if an increase in ‘a’ makes decrease in ‘b’ proportionally and vice versa. It can be expressed as a=^{k}⁄^{b}where k is called the constant of proportionality. Eg:- If I need to go a distance of S with velocity V and time T then T =^{S}⁄^{V}. Here the distance S is constant. If velocity increases it will take less time so T decreases. So T is in indirect variation or inverse relation with V.**Joint variation**: Given three variables a , b and c. C is said to be in joint - variation with ‘a’ and ‘b’ , if it varies with both variables. It can be expressed as c=kab , where k is the constant of proportionality.

Eg:- Men doing a work in some number of days working certain hours a day.

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