Today in "laws of nature" we are going to talk about the very important arithmetic formulae on AVERAGE , which are very helpful for all types of competitive examination.

AVERAGE PROBLEM SOLVING TRICKS

1). If the number of the quantities and it's sum is given then the average is given by = sum of the quantities/no. of the quantities

2). If the no. of the quantities and it's average is given then the sum of the quantities is = no. of the quantities × it's average.

3). If the sum and it's average is given then the the no. of the quantities is = sum of the quantities/ average

4). If the average of A boys is x , and the average of B girls is y, if all of them is put together , then average is = (Ax + By)/(A+B)

5). If average of A boys is x , and average of B boys out of A boys is y then average of rest boys is = (Ax-By)/(A-B)

6). If the average of A objects is x , if any object is removed from the list then average become y then the magnitude of removed object is =A(x-y)+y

7). If the average of A no. is a , if x is added or subtracted from each no. then the average become = (a+x) if added or (a-x) if subtracted.

8). If average of A quantities is x , and if a new quantity is added then average become y, then the added new quantity is =A(y-x)+y

9). If the average of A no. is x , then if each no. is multiplied or divided by y then , the average become = xy if multiplied or x/y if divided.

10). If the average weight of A person is increased by x kg if one person of weight y kg is replaced by a new person , then the weight of new person is = y + Ax
*If average is decreasing then the weight of new person is = y- Ax

11). The average marks obtained by A candidates in a certain examination is m, if the average marks of passed candidate is n, and the fail candidate is o, then the no of candidates who passed the exam is =
[A(m-o)/n-o] and no of candidates who failed the exam is =[A(n-m)/n-o]

12). If average of n no. is 'a' , (where n is odd no.) And average of first (n+1/2) no. is 'b' and
Last (n+1/2) no. is 'c' then the  (n+1/2)th no. is = [(n+1/2)(b+c) - na]

13). If a batsman in n innings makes a score of x , and average is increased by y, then average after n innings is = [x-y(n-1)]

14). If a batsman has average of x runs after the completion of n innings. Then no of runs he has to make to raise his average to y is =
[n(y-x)+y]

15). If a person travels a distance with x km/h , and again travels the same distance with y km/h, then average speed for whole journey is = 2xy/x+y km/h

* If half of the distance is travelled by x km/h and other half distance is travelled by y km/h then average for whole journey is =
  2xy/x+y km/h

* If person goes with x km/h and return with y km/h then average speed is = 2xy/x+y km/h

16). If a person travels three equal distances with x km/h , y km/h and z km/h then average speed for whole journey is =
         3xyz/xy+yz+zx km/h

17). If a person travels A km by x km/h, B km by y km/h and C km by z km/h then average speed for whole journey is =

18). A person travels Ath part of a distance with x km/h , Bth part of distance with y km/h, and Cth part with z km/h then average speed for whole journey is =
                                       1/(A/x +B/y + C/z). Km/h

* If Ath, Bth and Cth part of distance is given as A%, B% and C% then formula change to average speed of = 100/(A/x + B/y + C/z) km/h

19). The average value of all the members of a group is x, if the first part of members has average of y, and average of remaining parts of members is z and no. of members in first part is n then no. of members is remaining part =[n(x-y)/z-x]

* If n is the no. of members in remaining part then the no. of members in first part is =[n(x-z)/y-x]

20). The average of first n natural number is = n+1 /2

21). The average of n consecutive number is the middle no. ( Where n is odd number).

22). Average of n consecutive number is the average of middle two numbers. ( Where n is even number).

* The average of two middle number is calculated as follows:
** In case of consecutive numbers,
 Average = smaller middle no. +0.5 or greater middle number -0.5
** In case of consecutive odd and consecutive even.
  Average = smaller middle no. +1 or greater middle no. -1
   

23). The average of odd number from 1 to n, (where n is natural odd number) is =last odd number +1 /2

24). The average of even number from 1 to n, (where n is natural even number) is = last even number +2 /2

25). The average of square of natural number till n is = [(n+1)(2n+1)/6]

26). The average of cubes of natural number till n is = [n(n+1)^2/4]

27). The average of first n consecutive even number is =n+1

28). The average of first n consecutive odd number is =n

29). The average of squares of first n consecutive even number is =
                                                    [2(n+1)(2n+1)/3]

30). The average of squares of even number till n is =[(n+1)(n+2)/3]

31). The average of squares of consecutive odd number till n is =[n(n+2)/3]

32). The average of n numbers is A , and rechecking it is find that some of the numbers that is (x1 , x2, x3, ...xn) are taken wrongly as ( x1', x2', x3', ...xn') then the correct average is =   A + [( x1+x2+ x3 +...xn) - (x1'+ x2'+ x3' + ...xn')]/n

33). Average of a series having common difference 2 is = first term + last term /2

34). If the average of n consecutive odd numbers is x , then the difference between the largest and smallest number is = 2(n-1)

35). If P distance is travelled by x km/h, Q distance with y km/h, R distance with z km/h, then average speed for whole journey is = (P+Q+R)/(P/x + Q/y + R/z) km/h

36). The average weight of group of X members is y, if after entering or exiting of a member , average weight become z, then weight of entering or exiting person is =
                                                                  z+-x(z-y)

37). The average weight of group of x person is y, when z person get enter/exit in the group , the average of group become w, then average of new entering/exiting persons is =  y +-(x/z +1)w

38). In the group of x persons , if a t years old person is replaced by a new person , then average is increased/ decreased by t1 , then age of new person is = t+-xt1

39). The average of n multiple of any number is = no.(n+1)/2

These are all the important arithmetic formulae on AVERAGE...