This is a basic concept in arithmetics and it is important to solve many questions asked in various competitive exams that include averages of income, age, speed, time, distance, tabulation based or frequency based arithmetic means, increase or decrease in average.

## AVERAGE

The averages or arithmetic mean or mean of two or more quantities is equal to their sum divided by the number of those quantities.

\begin{align*} Average(Avg)=\dfrac {\left( Sum \ of \ quantities\right) }{\left( No. \ of \ quantities\right) } \end{align*}

\begin{align*} Average(Avg)=\dfrac {\left( x_{1}+x_{2}+x_{3}\right) }{3} \\ here \ x_{1}, x_{2}, x_{3} are \ quantities \end{align*}

**For example: **

average of numbers 10,20,50,40,16,28 is [(10+20+50+40+16+26)/6] = 27.

### Weighted Average (WA)

If the given quantities (x1,x2,x3) are occurring with a certain given weight (w1,w2,w3) then the average is a weighted average.\begin{align*}

WA =\dfrac {\left[ x_{1}\times w_{1}+x_{2}\times w_{2}+x_{3}\times w_{3}\right] }{w_{1}+w_{2}+w_{3}}

\end{align*}

### Formulas Of Average Concept

1. Average of two or more groups, when taken together, is calculated as

- Let us consider a number of quantities in two groups as a and b and their average is x and y respectively then
*combined Avg. of all of them taken together is*

\begin{align*} \dfrac {ax+by}{a+b} \end{align*} - Let us consider the averages of quantities is x and the average of b quantities out of them is y then
*Avg. of the remaining group is*\begin{align*} \dfrac {ax-by}{a-b} \end{align*}

2. The average of quantities is equal to x if one of the quantities (b) be replaced by a new quantity (c) the average becomes y then \begin{align*} c=b+a\left( y-x\right) \end{align*}

3. The average of quantities is equal to x, if one of the quantities is removed then the Avg. becomes y.*The value of the removed quantity is* \begin{align*} a\left( x-y\right) +y \end{align*}

4. The average of quantities is equal to x, if one of the quantities is added then the Avg. becomes y. *The value of added quantity is* \begin{align*} a\left( y-x\right) +y

\end{align*}

5. Let a person A goes from P to Q with a speed of x km/h and returns from Q to P with a speed of y km/h, then *the Avg. speed of total journey*

\begin{align*} \dfrac {2xy}{\left( x+y\right) }

\end{align*}

6. Let a distance d is traveled with three different speeds a km/h, b km/h, c km/h then *Avg speed of the total journey *

\begin{align*} \dfrac {3abc}{\left( ab+bc+ca\right) }km/hr \end{align*}

7. In three numbers if the first number is a time the second number and b times the third number and the Avg of all the three numbers is x, *then **the first number is*

\begin{align*} \dfrac {3ab}{\left( a+b+ab\right) }\times x \end{align*}

8. Let the average of n numbers is x later it was found that a was misread as b. Now *the correct Avg. will be is*

\begin{align*} m+\left[ \dfrac {\left( a-b\right) }{n}\right] \end{align*}

9. Let the average of n numbers is x later it was found that a and b were misread as c and d. Now *the correct Avg. will be is*

\begin{align*} m+\left[ \dfrac {\left( a+b-c-d\right) }{n}\right] \end{align*}

###### 10. Let Ā be the average of a1,a2,a3…….an then the average

- of a1+a,a2+a,a3+a…………an+a is
**Ā+a.** - for a1-a,a2-a,a3-a…………an-a is
**Ā-a.** - of a×a1,a×a2,a×a3………..a×an is
**a×Ā. here a≠0.** - of a1/a ,a2/a ,a3/a ………….an/a is
**Ā/a. here a≠0.**

11. The averages of the first n natural numbers is

\begin{align*} \dfrac {n+1}{2} \end{align*}

12. The Avg. of the sq. of 1st n natural numbers is

\begin{align*} \dfrac {\left( n+1\right) \left( 2n+1\right) }{6}\end{align*}

13. The Avg. of the cube of first n natural numbers is

\begin{align*} \dfrac {\left( n+1\right) ^{2}}{4} \end{align*}

14. The Avg. of first n consecutive even numbers is

\begin{align*} \ n+1 \end{align*}

15. The Avg. f first n consecutive odd numbers is **n.**

16. The Avg. of the square of first n consecutive even numbers is

\begin{align*} \dfrac {2\left( n+1\right) \left( 2n+1\right) }{3}\end{align*}

17. The Avg. of the square of consecutive even numbers up to n is

\begin{align*} \dfrac {\left[ \left( n+1\right) \left( n+2\right) \right] }{3}

\end{align*}

18. The Avg. of the square of consecutive odd numbers up to n is

\begin{align*} \dfrac {n\left( n+2\right) }{3} \end{align*}

19. The Avg. of even numbers from 1 to n is

\begin{align*}

\dfrac {\left(last \ even \ number+2\right)}{2}\end{align*}

20. The Avg. of odd numbers from 1 to n is

\begin{align*} \dfrac {\left(last \ odd \ number+1\right)}{2}\end{align*}

21.let n is even:

The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is usually

\begin{align*} the \ Avg. \ of \ middle \ two \ numbers\end{align*}22.let n is odd:

The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers will be

\begin{align*} the \ middle \ number\end{align*}