# Average Definition and formulas

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## Average Definition

Average in general means a number which represents the group of numbers is called the average number or in other words the sum (S) of all the given results/amounts/quantities/numbers when divided by the total numbers (n). The result obtained is called the average of the set of all those numbers.

$$Average (A) = \frac {sum\hspace{1mm}of \hspace{1mm}amounts/quantities/numbers \hspace{1mm}(s)}{Total \hspace{1mm} number \hspace{1mm}of \hspace{1mm}amounts/quantities/numbers \hspace{1mm}(n)}$$

Example1- Find the average of 3, 4, 5, 8, 9, 4?

Solution- n = 6

$$Average (A) = \frac {3+4+5+8+9+4}{6}$$

$$Average (A) = \frac {33}{6}$$

= 5.5

Example2- The ages of A, B and C are 32 years, 36 years and 46 years respectively, what is their average age?

Solution- n = 3

$$Average (A) = \frac {32+36+46}{3}$$

$$Average (A) = \frac {114}{3}$$

= 38

Example3- Find the average of 1, -4, 5, 6, 7, -9?

Solution- n = 6

$$Average (A) = \frac {1+(-4)+5+6+7+(-9)}{6}$$

$$Average (A) = \frac {1-4+5+6+7-9}{3}$$

$$Average (A) = \frac {6}{6}$$

= 1

### Some Important formulas

1. If P goes from A to B with a speed of x km/h and comes back from B to A with a speed of y km/h, then the average speed of the whole journey is,

Average = $$\frac {2xy}{x+y}$$ km/h

2. If P covers a distance from A to B at three different speeds respectively x km/hr, y km/hr and z km/hr then the average speed of the total journey,

Average Speed = $$\frac {2xyz}{xy+yz+zx}$$ km/h

3. If the average of 'm' numbers is 'x' and that of 'n' numbers out of these 'm' numbers is 'y', then the average of the remaining numbers will be-

(i) Average of remaining numbers = $$\frac {mx-ny}{m-n}$$ (If m > n)

(ii) Average of remaining numbers = $$\frac {ny-mx}{n-m}$$ (If m < n)

4. Average of first 'n' natural numbers,

$$= (1+2+3.................+n) = \frac {n+1}{2}$$

5. Average of first 'n' odd natural numbers,

= {1+3+5+..................+(2n-1)}= n

6. Average of first 'n' even natural numbers,

= {2+4+6+..................+2n} = n+1

7. Average of squares of first 'n' natural numbers,

= (1+ 2+ 3+ 4+ ........................+ n2) = $$\frac {(n+1)(2n + 1)}{6}$$

8. Average of cubes of first 'n' natural numbers,

= (1+ 2+ 3+ 4+ ........................+ n3) = $$\frac {n(n+1)^{2}}{4}$$

9. Average of first 'n' multiples of number 'x' = $$\frac {x(n+1)}{2}$$

10. If 'a' is the average of 'n' numbers and 'b' is the average of 'm' numbers, then the average of the total numbers 'n' and 'm' will be-

Average of the total numbers 'n' and 'm' = $$\frac {na+mb}{n+m}$$

Average Questions with solutions