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## 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