Ratio and Proportion | Introduction & Concept
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 a2:b2
- Triplicate Ratio: The triplicate ratio of a:b is a3:b3
- 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
- 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, b2 = 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 c2=ab , and c is called the mean proportional of a and b . Formulae of mean proportion is , in this case c = √ab
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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