Average

Average 

Average Concept:-

      The average of a number is a measure of the central tendency of a set of numbers.

       In other words, it is an estimate of where the center point of a set of numbers lies.

Average:-It is defined as the ratio of sum of observations to Number of observations.

          

Average =	Sum of observations
	Number of observations

Some Important Results:- 

(1) Average of n natural numbers = (n + 1) / 2

(2) Average of even n natural numbers = (n + 1)

(3) Average of odd n natural numbers = n
(4)  If value of each term increases/decreases by k, then the average of the group also

      increases/decreases by k. 

(5) If we know average of two groups individually, then the average of combined group 
      cannot be determined.
(6) There are two groups A and B .If A has n₁ observations B. has n₂ observations and
     their averages are x₁ and x₂ respectively. If we have to find the average of the 
    Combined group of A and B use the formula
      
     

Average of whole group =	(n1x1 + n2x2)
	(n1 + n2)

Example 1 :-Average of 4 consecutive number is 5.5 find the largest number.

                   Let numbers are x,x+1,x+2,x+3.
 Explanation     Average =  x+(x+1)+(x+2)+(x+3)/4 = 5.5

                    4x + 6 = 22

                    4x = 16

                     x = 4

                   largest number is x+3 = 4+3 = 7

Example 2 :-Average  of 4 consecutive odd number is 24 find the smallest number.
 Explanation    Let numbers are x,x+2,x+4,x+6.

                   Average =  x+(x+2)+(x+4)+(x+6)/4 = 24

                    4x + 12 = 96

                    4x = 84

                     x = 21

                   Smallest number = x = 21

Example 3 :-The average age o three boys is 15 years.If their age are in ratio 3:5:7
                      what is the age o youngest boy.
 Explanation    Let the ages of three boys be 3x,x,7x
                      Average age =(3x+5x+7x)/3 = 15
                       15x=45
                        x=3
                       The age o youngest boy is 3x=3*3=9

Example 4 :-There were 35 students in a hostel. due to the admission 

                      of 7 new students the expenses for the mess were increased 

                      by 42/- per day while the average expenditure per head diminished

                      by Rs 1 .what was the original expenditure of the mess.
 Explanation    After the  7 new students, number of students will become 42

                     average expenditure per head = (x-1)

                     New total expenditure = 42(x-1)

                     Given that Expenses for the mess were increased by 42

                     42(x-1) -35x =84

                   42x-42-35 =84

                   7x = 84

                   x=12
Example 5 :-There are two sections A and B of a class, consisting of 36 and 44 students’ 
                    respectively.If the average weight of section A is 40kg and the average weight
                   of section B is 35kg, find the average weight of the whole class.
Explanation     Now n₁=36,n₂=44,x₁=40,x₂=35
                 Average of whole group =(n₁x₁ + n₂x₂)/(n₁ + n₂)
                                                          = (36*40 + 44*35)/(36+44)
                                                         =2980/80
                                                         = 37.25
Example 6 :-In the first 10 overs of a cricket game, the run rate was only 3.2

                     What should be the run rate in the remaining 40 overs to reach

                    the target of 282  runs.

Explanation   the first 10 overs  the run  was  32

                   the first 50 over’s  the run  was  282

                    the  remaining 40 overs  the run  was  282-32 = 250

                    the run rate= 250/40 = 6.2
Example 7 :-The average of 25 results is 18.The average of first twelve of them is 14
                        and that of last  twelve is 17.Find the thirteenth result.

Explanation 

1_2_3_4_5_6_7_8_9_10_11_12_13_14_15_16_17_18_19_20_21_22_23_24_25

                       Sum of 25 results is 18*25 = 450
                       Sum of first twelve results is 14*12 = 168
                       Sum of last twelve results is 17*12 = 204
                       Thirteenth result = Sum of 25 results
                                                        - (Sum of first twelve results + Sum of last twelve results)
                                                     = 450-(168 + 204)
                                                     =  450 - 372 = 78
Example 8:-The average of  11 results is 60.The average of first six result  is 58
                        and that of last  six is 63.Find the seventh result.
Explanation   1_2_3_4_5_6_7_8_9_10_11

Sum of 11 results is 11*60 = 660
                       Sum of first six results is 6*58 = 348
                       Sum of last six results is 6*63 = 378
                       Seventh result = (Sum of first six results +Sum of last six results)
                                                  -Sum of 11 results   
                                               = (348 + 378) - 660
                                               =  726 - 660 = 66