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…