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Ratio & Proportion

 

Ratio & Proportion

Ratio : The representation of same time of quantities in the simplest fraction.

Example : Ratio between marks of Girls and marks of Boys. 

                  Ratio between the Height of 1st and 2nd Person.

  • The simplest fraction x/y is represented as x:y.

Here x is called the Antecedent and y is called the Consequent.


Different types of Ratios are :

  1. Duplicate Ratio : It is the ratio of squares of two numbers.

                                                                       

Duplicate Ratio of the fraction X/Y is given as :

X/Y = X² / Y² or

                     X : Y = X² : Y²                                                 

  1. Sub - Duplicate Ratio : It is the ratio between square roots of two numbers.

                                                                    

Duplicate Ratio of the fraction X/Y is given as :

X/Y = √X /√Y

X : Y =√X : √Y      

  1. Triplicate Ratio : It is the ratio of cubes of two numbers.

                                                                    

Triplicate Ratio of the fraction X/Y is given as :

X/Y = X³ / Y³ or

             or,    X : Y = X³ : Y³   

  1. Sub - Triplicate Ratio : It is the ratio between the cube roots of two numbers.

                                                                   

Triplicate Ratio of the fraction X/Y  is given as :

X/Y = X⁽¹/³⁾ / Y⁽¹/³⁾

            or,     X : Y = X⁽¹/³⁾ : Y⁽¹/³⁾

  1. Compound Ratio : It is the ratio of product of first terms in every ratio to that of the second term in every ratio. 

Compound ratio of (a : x), (b : y) , (c : z) is (abc : xyz)



Properties of Ratio

Compounded Ratio 

Formulas

Addendo

If a : b = c : d = e : f = ...

(a + c + e + ….) = (b + d + f…..)

Subtrahendo

If a : b = c : d = e : f = ...

(a - c - e - ….) = (b - d - f…..)


  1. Inverse Ratio : The ratio formed by interchanging their old places in the ratio of new. 

The inverse ratio of 8 : 5 is 5 : 8.


Proportion : The ratio of two ratios is represented as Proportion. The equality of two ratios is called Proportion. 

When a : b = c : d is represented as a : b ::c : d, then a, b, c, d are said to be in proportion.

In (a : b ::c : d) , a & d are called the Extremes and b & c are called the Means.

              Product of Means = Product of Extremes

a x d         =          b x c

Different types of Proportion are :

  1. Mean Proportion : Mean Proportion between X and Y is √XY.

  2. Third Proportion : If A : B = B : C, then C is called as third proportion to A and B.

  3. Fourth Proportion :  If M : N = P : Q , then Q is called the Fourth Proportion to M, N and P.


Properties of Proportion


Properties of proportion

Fourth Proportion

Formulas

Simple 

If a : b = c : d

ad = bc

Invertendo 

If a : b = c : d

                          

b : a = d : c or 

b/a = d/c              

Alternendo

If a : b = c : d

                         

a : c = b : d or 

a/c = b/d

Componendo

If a : b = c : d

a + b         c + d

━━━  =  ━━━

     b              d

Dividendo 

If a : b = c : d

a - b           c - d

━━━ =   ━━━

    b              d

Componendo & Dividendo

If a : b = c : d

a + b         c + d

━━━  =  ━━━

  a - b         c - d






KEY HIGHLIGHTS :

  • If x = ky for some constant k, then we can say that x is directly proportional to y.

  •  If xy =k for some constant k, then we can say that x is inversely proportional to y.

  • If a number “K” is split into 2 parts - m and n

                        K * m       

First Part = ━━━━

                      (m + n)

                           K * n  

Second Part = ━━━━

                                     (m + n)

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