Average problems tricks | average arithmetic formulae | average short-cuts

Average problems tricks | average arithmetic formulae | average short-cuts

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We are going to talk about the very important arithmetic formulae on average, which are very helpful for all types of competitive exams.

AVERAGE PROBLEM-SOLVING TRICKS

Average problems tricks | average arithmetic formulae | average short-cuts

1). If the number of the quantities and their sum is given, the average is given by =

    \[\frac{\text{sum of the quantities}}{\text{no. of the quantities}}\]

2). If the no. of the quantities and its average is given then the sum of the quantities is =

    \[\text{no. of the quantities} \times \text{its average}\]

3). If the sum and its average is given then the no. of the quantities is =

    \[\frac{\text{sum of the quantities}}{\text{average}}\]

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 the average is =

    \[\frac{(Ax + By)}{(A+B)}\]

5). If the average of A boys is x, and the average of B-boys out of A boys is y then the average of the rest of boys is =

    \[\frac{(Ax-By)}{(A-B)}\]

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 the 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 becomes =

    \[(a+x)\; \text{if added or}\; (a-x)\; \text{is subtracted}\]

8). If the 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 becomes =

    \[xy\; \text{if multiplied or}\; \frac{x}{y}\; \text{if divided}\]

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 the new person is =

    \[y + Ax\]

*If the average is decreasing then the weight of the new person is =

    \[y- Ax\]

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

    \[\frac{A(m-o)}{(n-o)}\]

    \[\text{and no of candidates who failed the exam is} =\frac{A(n-m)}{(n-o)}\]

12). If the average of n odd numbers is ‘a’, and the average of first \displaystyle{\left(frac{n+1}{2}\right)} number is ‘b’ if last \displaystyle{\left(frac{n+1}{2}\right)} number is ‘c’ then the \displaystyle{\left(frac{n+1}{2}\right)}th no. is =

    \[\left[\left(\frac{n+1}{2}\right)(b+c) - na\right]\]

13). If a batsman in n innings makes a score of x, and the average is increased by y, then average after n innings is =

    \[[x-y(n-1)]\]

14). If a batsman has an 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 the average speed for the whole journey is =

    \[\frac{2xy}{x+y}\;km h^{-1}\]

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

    \[\frac{2xy}{x+y}\;km h^{-1}\]

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

    \[\frac{2xy}{x+y}\;km h^{-1}\]

16). If a person travels three equal distances with x km/h, y km/h, and z km/h then the average speed for the whole journey is =

    \[\frac{3xyz}{xy+yz+zx}\;km h^{-1}\]

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 the average speed for the whole journey is =

    \[\frac{A+B+C}{\frac{A}{x}+\frac{B}{y}+\frac{C}{z}}\;km h^{-1}\]

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

    \[\frac{1}{\left(\frac{A}{x}+\frac{B}{y}+\frac{C}{z}\right)}\;km h^{-1}\]

* If Ath, Bth, and Cth part of the distance is given as A%, B%, and C% then formula change to the average speed of =

    \[\frac{100}{\left(\frac{A}{x}+\frac{B}{y}+\frac{C}{z}\right)}\;km h^{-1}\]

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

    \[\frac{n(x-y)}{(z-x)}\]

* If n is the no. of members in the remaining part then the no. of members in the first part is =

    \[\frac{n(x-z)}{(y-x)}\]

20). The average of first n natural number is =

    \[\frac{n+1}{2}\]

21). The average of n consecutive number is the middle number

22). The average of even n consecutive numbers is the average of the middle two numbers.

* The average of two middle number is calculated as follows:
** In case of consecutive numbers-
Average = smaller middle number +0.5 or greater middle number -0.5

** In case of consecutive odd and consecutive even-
Average = smaller middle number +1 or greater middle number -1

23). The average of odd number from 1 to n, (where n is a natural odd number) is =

    \[\frac{\text{last odd number} +1 }{2}\]

24). The average of even natural number from 1 to n is =

    \[\frac{\text{last odd number} +2}{2}\]

25). The average of the square of natural number till n is =

    \[\frac{(n+1)(2n+1)}{6}\]

26). The average of cubes of natural number till n is =

    \[\frac{n(n+1)^2}{4}\]

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

    \[n+1\]

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

29). The average of squares of first n consecutive even number is =

    \[\frac{2(n+1)(2n+1)}{3}\]

30). The average of squares of even number till n is =

    \[\frac{(n+1)(n+2)}{3}\]

31). The average of squares of consecutive odd number till n is =

    \[\frac{n(n+2)}{3}\]

32). The average of n numbers is A , and rechecking it is find that some of the numbers that is (x_1 , x_2, x_3, ...x_n) are taken wrongly as (x_1', x_2', x_3', ...x_n') then the correct average is =

    \[A + \frac{\left[\left( x_1+x_2+ x_3 +...+x_n\right) - \left(x_1'+ x_2'+ x_3' + ...+x_n'\right)\right]}{n}\]

33). The average of a series having a common difference 2 is =

    \[\frac{\text{first term} + \text{last term}}{2}\]

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 =

    \[\frac{P+Q+R}{\frac{P}{x}+\frac{Q}{y}+\frac{R}{z}}\;km h^{-1}\]

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

    \[z\pm x(z-y)\]

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

    \[yy\pm\left(\frac{x}{z} +1\right)w\]

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

    \[t\pm xt_1\]

39). The average of n multiple of any number is =

    \[\text{number}\times \frac{(n+1)}{2}\]

For more such tricks articles stay tuned with Laws Of Nature

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