Important Notes : Ratio And Proportion

RATIO AND PROPORTIONS 

Ratio: Ratio is the comparison of two quantities. The ratio of ‘A’ to ‘B’ is written as

                        A : B = 2 : 3

                  𝑨/𝑩  = 𝟐/𝟑

CONCEPT OF PROPORTION


Proportion:  A proportion is an expression which states that two ratios are equal.

  =    is a proportion

TYPE OF PROPORTION

If  A, B, and C are in proportion.

 e.g.

            A : B :: B : C
           

MEAN PROPORTION 


B²  =  
  B   =  

THIRD PROPORTION 

FOURTH PROPORTION 

If four quantities A, B, C and D are in proportion 

                     So,            A : B : : C : D 

                                          
                             

Ex:    If A and B are in the ratio 4:5 and the difference of their squares is 81, what is the value of A.

Sol:  


 x = 3
So A = 3 × 4 = 12

Ex:    Two alloys contain tin and iron in the ratio of 1:2 and 2:3. If the two alloys are mixed in the proportion of 3:4 respectively (by weight), the ratio of tin and iron in the newly formed alloys is.

          Ist Alloy suppose 3× 3 = 9kg and IInd alloy  5 × 4 = 20
               3  6               
          8        12             
          11   : 22
                    1  :   2

 Ex: In a bag 240 coins of 50 paise, 25 paise and 10 paise and their value are in the 5 :3 : 1 then finds the number of 25 paise coins.



                    = 5 : 6 : 5
25 paise coins =  × 6 = 90

Ex: Price of a piece of a diamond changes according to the square of its weight. If a diamonds of Rs 5184 is cut in three parts in such a way that weight of parts in such a way that weight of these parts in the ratio 1 : 2 : 3 , then find the amount of loss by cutting the diamond.



x2 = 5184
x = 72
Ratio of the part of the diamond = 1: 2 : 3
Weight of parts   = 12 , 24 , 36
Total value = 122 + 242 + 362  
                   = 2016
Loss = 5184 – 2016 = 3168

Ex: Three persons A, B, C whose salaries together amount to Rs 72000 spend 80, 85 and 75 percent of their salaries respectively. If their savings are in the ratio 8 : 9 : 20, then A's salaries is-

Sol:  
If the salaries of A, B and C be Rs. x, Rs. y and Rs. z respectively then

x:y:z = 40 : 60 : 80
= 2 : 3 : 4
A's salary =
Rs.

Ex:    The ratio of the first and second class fares between two railway stations is 4 : 1 and that of the number of passengers travelling by first and second classes is 1 : 40. If on a day Rs. 1100 are collected as total fare, the amount collected from the first class passenger is.
Sol:   Ratio of the first and second class fares (total)
=1× 4 : 1 × 40
= 4 : 40 = 1 : 10

• Amount collected from the first class passengers

        
Ex:    What least number must be subtracted from each of the numbers 14, 17 , 34 and 42 so that the remainders may be proportional?

          Sol:   Let the required number be x.
                   Then, (14 – x) : (17 – x) :: (34 – x) : (42 – x)
                   
                   
                    x2 – 56x + 588 = x2 – 51x + 578
                    5x = 10 
                   Required number = 2

Ex:  Seats for Mathematics, Physics, and Biology in a college are in the ratio of 5 : 7 : 8 . There is a proposal to increase these seats by 40%, 50%, and 75% respectively.What will be the ratio of seats after increase?
Sol:
          Initial ratio = 5 : 7 : 8

     After increase = 5  : 7 : 8

                              = 2 : 3 : 4

Ex:   A, B and C have 40, X and Y stones with them respectively. If B gives 20 stones to A , he is left with half as many stones as C. If together they had 60 more stones, each of them would have had 100 stones on an average. What is the value of X : Y ? 
Sol :
               =100
                    X + Y = 200 -------(1)
                       = 
                   2X – Y = 40 --------(2)
   From both the eq.
      X = 80 , Y = 120
             X : Y = 2 : 3