Average

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1. Average → The average of N numbers is their sum divided by N, that is

Average = fraction numerator Sum space of space straight N space given space number over denominator straight N end fraction

2. Weighted average → The concept of weighted average is used mostly when different sets of numbers is given.

As 3 section of a class, having strength of a, b and c have their average marks x, y and z.

So average marks of whole class = fraction numerator ax space plus space by space plus space cz over denominator straight a space plus space straight b space plus space straight c space end fraction

The coined term is called weighted average.

Note → The average of two sets of numbers is closer to the set with more number.

3. Average speed → Suppose a man covers a distance with x km/hr and equal distance with y km/hr, then average speed during the whole journey is = fraction numerator 2 xy over denominator straight x space plus space straight y end fraction

And if his total distance is divided in 3 parts each covering with x, y and z km/hr then average speed = fraction numerator 3 space xyz over denominator xy space plus space yz space plus space zx end fraction

Some tricks for average →

1) If some given number have their average and.

(a) If each number is increased by a, then new average = old average + a

(b) If each number is decreased by a, then new average = old average – a

(c) If each number is multiplied by a, then new average = a × old average

(d) If each number is divided by a, then new average = old average / a

2) If average of n1 numbers is x1, and of n2 numbers is x2 then average of all numbers = fraction numerator straight n subscript 1 straight x subscript 1 space plus space straight n subscript 2 straight x subscript 2 over denominator straight n subscript 1 space plus space straight n subscript 2 end fraction

3) If average of 'm' numbers is 'x' and out of 'n' numbers have their average 'y', then average of remaining numbers = fraction numerator mx space – space ny over denominator straight m space – space straight n end fraction

4. Some tricks for natural numbers →

(a) average of first 'n' natural numbers = open parentheses fraction numerator straight n space plus space 1 over denominator 2 end fraction close parentheses

(b) average of first 'n' even numbers which are natural = n + 1

(c) average of first 'n' natural odd numbers = n

(d) average of 'n' consecutive numbers = fraction numerator first space no. space plus space last space no. over denominator 2 end fraction

(e) average of square of first 'n' natural numbers = fraction numerator left parenthesis straight n space plus space 1 right parenthesis space left parenthesis 2 straight n space plus space 1 right parenthesis over denominator 6 end fraction

(f) average of cubes of first 'n' natural numbers = fraction numerator straight n space left parenthesis straight n space plus space 1 right parenthesis squared over denominator 4 end fraction

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