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Average


Average : It is defined as the ratio of Sum of all the Observations to the Number of Observations. 

Average = Sum of all the observations / No. of Observations

                                      

OR, Sum of all the observations = Average x No. of Observations


To Calculate the Average Speed of the journey To and Fro 

Going from A to B with the speed of  X kmph; 


Coming from B to A  with the speed of Y kmph.


                                   A                        X kmph  →                                  B

                                   ⧭━━━━━━━━━━━━━━━━━━━━━⧭

                                                                 ←  Y kmph


                                                                       

    Average Speed during whole journey = 2(X * Y) /(X + Y)

                                                                       


To calculate the Average of the whole class

Where,  Slot A = Number of students = X

                           Average of Slot A     = M

and ,     Slot B = Number of students  = Y

                           Average of Slot B     = N


                                               XM + YN

 Average of whole Class =   ━━━━━     

                                                  X + Y


Weighted Average : 

                                       W₁X₁ + W₂X₂ + W₃X₃

Weighted Average  =  ━━━━━━━━━━━━

                                          W₁ + W₂ + W₃


Here, X₁ , X₂, X₃ and X₄ are the respective averages of W₁, W₂, W₃ and W₄.



Problems Based on Age :

When New Member is Added in the group

Case 1 : Average Age Increases

Age of New Member added = Previously given Age + ( Increase in average after new member added * Total members including new member )

Case 2 : Average Age Decreases

Age of New Member added = Previously given Age - ( Decrease in average after new member added * Total members including new member )


When a member of group is replaced by the another new member in the group

Case 1 : Average Age Increases

Age of New Member added = Age of Departed Member + ( Increase in average after new member added * Total members including new member )

Case 2 : Average Age Decreases

Age of New Member added = Age of Departed Member  - ( Decrease in average after new member added * Total members including new member )


Average of Series : 


Series 

Sum

Average

(1 + 2 + 3 + 4 + ………….n)

½ {n (n+1 )}

½  (n+1 )

( 1² + 2² + 3² + ………….n²)

⅙ { n ( n + 1) ( 2n + 1)}

⅙ { ( n + 1) ( 2n + 1)}

( 1³ + 2³ + 3³ + ………….n³)

[½ {n (n+1 )}]²


n (n+1 )²

━━━━

4

(1 + 3  + 5 + 7 + ............. )

n²      ( n = no. of terms)

n      ( n = no. of terms)

( 2 + 4 + 6 + 8 + .................)

  n (n + 1 )


  (n + 1 )



KEY HIGHLIGHTS : 

                                                                                       

  • Average of N consecutive natural numbers

  • First + Last

  • =  ━━━━━━━ 

           2                                                                     

  • The Average of Odd no. of Observation in consecutive pattern  = MIDDLE TERM

Example : Calculate the average of  4, 8, 12, 16, 20

Sol : Average = Middle term = 12

  • The Average of Even no. of Observation in consecutive pattern  = Average of Middle two terms

Example : Calculate the Average of 6, 12, 18, 24, 30, 36, 42, 48

Sol : No. of terms = 8 (EVEN) 

                                           24 + 30               54 

                   Average =   ━━━━━━  =   ━━━ =  27

                                                            2                     2    

  • If each term of the observation is increased or decreased by K, the average eventually increases or decreases by K.

  • If the average of the two separate groups is known, then the combined average of the group can not be determined.

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