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                                                 

2. 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      

3. 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   

4. 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⁾

5. 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…..)

6. 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)