Short Tricks On Averages

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Average

1 What is Average?

A calculated "central" value of a set of numbers.

Average =          Sum of Observation

                         Number of Observation

When we cross multiply

Sum of observation = Average of Observations * Number of observation

           

2 In case of Average speed, the formula will change

What is Average Speed?

Average Speed is the total distance traveled by an objects divided by total time taken.

If distance and time are not mentioned rather the speeds are given, then;

Suppose a man covers a certain distance at X km/h and an equal distance at Y km/h

Then the Average Speed during the whole journey is

Average Speed =   2XY

                              X+Y

When there are three speeds X km/h, Y km/h & Z km/h

Average Speed =   3XYZ

                             XY+YZ +ZX

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3. If the average of n1 items is p1 and the average of n2 items is p2 then

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The average of total items will be

n1p1 + n2p2  

    n1 +n2

Example: There are 60 students in a class. Out of which 40 are boys and 20 are girls. If the average age of boys is 15 years and that of girls is 16 years then find the average age of the class.

40*15 + 20*16.

          60

Ans. 15.33 years

4. When we have to find the average of consecutive odd numbers

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Then,

Case 1: - When the number of observation is Even

Example: - 11, 13, 15, 17 (here the number of observation is 4, even)

Divide the numbers equally in two parts

11, 13 …….15, 17

The number between middle two terms is average, i.e. 14 in this case.

Alternative method: Sum of extremes divided by 2

Here 11+17 = 14

                2

Case 2: - When the number of observation is Odd

Example: - 163, 165, 167,169,171 (here the number of observation is 5, Odd)

Here the middle term is the average i.e. 167

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5. Average of consecutive first n natural numbers is

    n+1    .

      2

Average of first 20 natural numbers is 20+1 divided by 2 i.e. 10.5.

Sum of first consecutive n natural numbers is

n(n+1)  .

    2

As

Sum of Observations = Average of Observations * Number of observation

6. Average of consecutive first n natural even numbers is n+1.

Example: - Average of consecutive first 5 natural even numbers is 5+1 =6

Long method      2+4+6+8+10

                                              5

30/5= 6

Average of consecutive first n natural odd numbers is n.

Example: - Average of consecutive first 5 natural odd numbers is 5

Long method      1+3+5+7+9

                                              5

25/5= 5

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Average of n multiples of p will be

 

    p(n+1)  .

         2

Example: Average of 3 multiples of 5 is

5(3+1)/2 =

Long method 5+10+15

                                    2

Ans. 10

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