# Average: Basic Concepts, Formulas

## Definition-

Topics Discussed:

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,

The sum of the quantities = Average × the number of the quantities

## Average Formulas

• 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-

$(\frac { 2xy }{ x+y } )\quad km/hr$

• 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-

$(\frac { 3xyz }{ xy+yz+zx } )\quad km/hr$

• Average of first ‘n’ natural number is-

$(\frac { n+1 }{ 2 } )$

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

$\frac { n{ (n+1) }^{ 2 } }{ 4 }$

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

$[\frac { (n+1)(2n+1) }{ 6 } ]$

Profit and Loss Problems (MCQs)

## 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. The 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 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

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

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