**Important Facts and Formulas- Average**

**Definition-**

The number of quantities of the same kind is their sum divided by the number of quantities.

*In short,*

**Average = (Sum of the Quantities / Number of the quantities)**

*Also*,

**Sum of the quantities = Average × the number of the quantities**

- If a person covers a distance at x km/hr, and he again covers the same distance at a speed of y km/hr. Then his Average speed is –

**[2xy/(x + y)] km/hour.**

- If a person covers three equal distances at a speed of x km/hr, y km/hr, and z km/hr. Then the average speed during the whole journey is –

**[3xyz/(xy + yz + zx)] km/hr**

- Average of first ‘n’ natural number is –

**(n + 1)/2**

- Average of cubes of first ‘n’ natural numbers is –

**[n(n + 1 ) ^{2}/4]**

- Average of squares first ‘n’ natural number is –

**[(n + 1)(2n + 1)/6]**

__Examples__

**Q- 1> Find the average of first 10 multiples of 7.**

**Solution-**

Required Numbers = 7(1+2+3+4+5+6+7+8+9+10)

Here n = 10

Using Formula –

Average = (n + 1)/2

7(10 + 1)/2 = 77/2 = 38.5 (Ans.)

**Q- 2> Distance between Point A and Point B is 778 km. A car covers the distance from A to B at 84 km/hr and returns back to A with a uniform speed of 56 km/hr. Find the average speed of the car during the whole journey.**

**Solution-**

Let x = 84 km/hr, y = 56 km/hr

*Using the Formula*

Average Speed = [2xy/(x + y)] km/hr

= [(2 × 84 × 56)/(84 + 56)] = [(2 × 84 × 56) /140] = 67.2 km/hr