Ratio & Proportion

Ratio:-Ratio is a comparison between two quantities in same unit by using division.

              Ratio of two values x and y is written as x:y or x/y or x to y.

Type  of ratio:-

(1).Duplicate ratio:-The ratio of square of two numbers is called duplicate ratio.

                                 The duplicate ratio of x and y is x²/y²_.

(2)Triplicate ratio:-The ratio of cube of two numbers is called triplicate ratio.

                                The triplicate ratio of x and y is x³/y³_.

(3).Sub duplicate ratio:-The ratio of square root of two numbers is called sub duplicate ratio.

                                      The sub duplicate ratio of x and y is √x/√y_.

(4) Sub triplicate ratio:-The ratio of cube root of two numbers is called sub triplicate ratio.

                                      The sub triplicate ratio of x and y is ∛x/∛y_.

(5) Inverse ratio:-The ratio between the reciprocals of two quantities is called inverse ratio.

                            The inverse ratio of x and y is y/x.

(6) Compound ratio:-The product of two or more ratios; thus xz:yw is a ratio compounded 

                                   of the simple ratios x:y and z:w.

Proportion:-Proportion is the equality of two ratio.

                         If a:b=c:d we write a:b::c:d.

                         Product of means = Product of extremes

                         ⟹ad = bc

Type  of Proportion:-

(1).Fourth Proportional:-If a:b = c:d then d is called fourth proportional.

(2).Third Proportional:-If a:b = b:c then b is called third proportional.

(3).Mean Proportional:-If a:x = x:b then x is called third proportional.

                                      It can also written as x =√(ab)

(3).Componendo and Dividendo rule:-If a:b = c:d

                                       Then (a+b)/(a-b)=(c+d)/(c-d) is componendo and dividendo rule.

Example 1.:-If a:b = 5:7 and b:c = 6:11 then find a:b:c.
                    
Explanation:-   a       :        b          :            c

                         5                7
                                           6                       11

                        a       :         b          :            c

                        5*6            7*6
                                          6*7                   11*7
                         30             42                     77
                        Hence  a:b:c = 30:42:77
Example 2.:-Find the fourth  proportional to the numbers 60,48,30.
Explanation:- Let fourth proportional is x.
                       ⟹ 60:48::30:x
                       As  Product of means = Product of extremes
                       ⟹ 60*x = 48*30
                       ⟹ x = 48*30/60 = 24
                      Hence fourth proportional is 24.
Example 3.:-Find the third  proportional to the numbers 16,36.  
Explanation:- Let third proportional is x.
                      Thus   16:36::36:x
                      As  Product of means = Product of extremes
                       ⟹ 16*x = 36*36
                       ⟹ x = 36*36/16 = 81
                      Hence third proportional is 81.    
Example 4.:-If 76 is divided into four parts proportional to 7,5,3,4.
                       Then find the smallest part.  
Explanation:- Given ratio =7:5:3:4.
                      Sum f the ratio is 19
                      Smallest part is (3/19)*76
                                                 = 12
                      Hence smallest part is 12.
Example 5.:-A bag contain 50 paise,25 paise,10 paise coins in the ratio 5:9:4.
                       Amounting to be Rs. 206 Find the number of coin of 50 paise.  
Explanation:- Let the number of 50 paise,25 paise,10 paise coins be 5x:9x:4x.
                       We know value of 50 paise coin is Rs. 0.5
                                        value of 25 paise coin is Rs. 0.25
                                        value of 10 paise coin is Rs. 0.1
                       Then 5x*0.5 + 9x*0.25 + 4x*0.1 = 206
                             2.5x + 2.25x + 0.4x = 206
                             5.15x = 206
                                    x = 40
                      Hence the number of coin of 50 paise is 5*40 = 200 
Example 6.:-The present ratio of ages of ages of two brothers is 4:5.
                      18 years ago their ages ratio was 11:16.Find the age of elder brother.  
Explanation:- Let the present ages are 4x:5x
                      Now given  

                      (4x-18)/(5x-18) = 11/16
                       64x-288 = 55x-198
                       9x = 90
                        x = 10

                       Hence the present age of elder brother is 5x = 5*10 = 50.