Average - Arithmetic

Rules to calculate average

Average of given numbers is equal to the sum of given numbers divided by the total number of numbers

Or, Average of given observations is equal to the sum of total observations divided by the number of observations.

Rule (1) Average of given numbers

Rule (2) Or, Average given observations

Rule (3) Calculation of average speed

Let a person covers a distance at a speed of km/h and again cover the same distance at a speed of y km/h.

Then average speed = ^{2 x y}⁄_{x + y} km/h

(4) Sum of n natural numbers

(5) Series of numbers in which the difference between two consecutive terms is equal is called Arithmetic series.

(a) Sum of n terms (S_n) of Arithmetic Series

= ⁿ⁄₂ [2 a + (n – 1) d]

Where, n = numbers of terms of arithmetic series

a = First terms of arithmetic series

And, d = difference between two consecutive terms, i.e. common difference of arithmetic series

(b) Sum of n terms (S_n) of Arithmetic Series

= ⁿ⁄₂ ( a + l )

Where, n = number of terms of arithmetic series

a = first term of arithmetic series

And, l = last term of arithmetic series

(c) t_n = a + (n – 1) d

Where, n = number of term of arithmetic series

a = first term of arithmetic series

d = common difference of arithmetic series

And, t_n = nth term of Arithmetic series

(1) Find the average of 1, 2 and 3

Solution :

We know that,

Average of given numbers

Here sum of given numbers = 1 + 2 + 3 = 6

And total numbers of given numbers = 3

Thus, the Average of given numbers = ⁶⁄₃ = 2

Thus, the Average of given numbers = 2 Answer

(2) Find the average of 1, 2, 3, 4 and 5

Solution:

We know that average of given numbers

Sum of given numbers = 1 + 2 + 3 + 4 + 5 = 15

And, the total number of given numbers = 5

Thus, Average = ¹⁵⁄₅ = 3

Thus, the average of given numbers = 3 Answer

(3) Find the average of 15, 16, 17, 20 and 12

Solution :

We know that, Average of given numbers

Thus, Average of given numbers = ^{15 + 16 + 17 + 20 + 12}⁄₅

= ⁸⁰⁄₅ = 16

Thus, Average of given numbers = 16 Answer

(4) Find the average of first 10 natural numbers.

Solution :

We know that, Counting numbers are called natural numbers.

Thus, first 10 natural numbers are

1, 2, 3 , 4, 5, 6, 7, 8, 9, 10

Now, We know that, Average of given numbers

Thus, average = ^{1+2+3+4+5+6+7+8+9+10}⁄₁₀

= ⁵⁵⁄₁₀ = 5.5

Thus, average of first 10 natural numbers = 5.5 Answer

Alternate method

We know that, Average of given numbers

We know that, sum of first n natural numbers = ^{n ( n + 1 )}⁄₂

Thus, sum of first 10 natural numbers

Here, n = 10

Thus, sum of first 10 natural numbers = ^{10 (10 + 1)}⁄₂

= ¹¹⁰⁄₂ = 55

And total numbers = 10

Thus, average of first 10 natural numbers = ⁵⁵⁄₁₀   = 5.5

Thus, average of first 10 natural numbers = 5.5 Answer

Shortcut method

We know that, Average of given numbers

And, sum of first n natural numbers = ^{n (n + 1)}⁄₂

Thus, Average of first 10 natural numbers = ¹⁄₁₀ × ^{10 (10 + 1)}⁄₂

= ¹⁄₁₀ × ¹¹⁰⁄₂ = 5.5

Thus, average of first 10 natural numbers = 5.5 Answer

(5) What will be the average of the first 25 natural numbers?

Solution :

We know that, Counting numbers are called natural numbers.

Thus, first 25 natural numbers are

1, 2, 3, 4, . . . . . . . , 25

We know that, Average of given numbers

[Sum of the first 25 natural numbers can be found by adding them. But it is easy to get their sum using the formula ]

We know that, sum of first n natural numbers = ^{n (n + 1)}⁄₂

Thus, sum of first 25 natural numbers = ^{25 (25 + 1)}⁄₂

= ^{25 × 26}⁄₂ = 325

Here sum of first 25 natural numbers = 325

And number of given numbers = 25

Thus, average = ³²⁵⁄₂₅   = 13

∴ Average of first 25 natural numbers = 13 Answer

(6) Calculate the average of first 50 natural numbers

Solution :

We know that, counting numbers are called natural numbers.

Thus, the list of first 50 natural numbers is

1, 2, 3, 4, 5, . . . . . . 50

[Sum of the first 50 natural numbers can be found by adding them. But it is easy to get the sum of the first 50 natural numbers using the formula ]

We know that, sum of first n natural numbers is ^{n (n + 1)}⁄₂

Therefore, sum of first 50 natural = ^{50 (50 + 1)}⁄₂

= ^{50 × 51}⁄₂

= ²⁵⁵⁰⁄₂ = 1275

Now, we know that average

Thus, average of given numbers = ¹²⁷⁵⁄₅₀ = 25.5

Therefore, average of given numbers = 25.5 Answer

(7) Find the average of first 5 multiples of 4.

Solution

List of first 5 multiples of 4 is

4, 8, 12, 16, 20

We know that average

= ^4+8+12+16+20⁄₅

= ⁶⁰⁄₅  = 12

Thus, Average of first 5 multiples of 4 = 12 Answer

(8) Find the average of first 10 multiples of 6.

Solution

List of first 10 multiples of 6

= 6 ( 1, 2, 3, . . . . , 10 )

Here numbers in bracket are natural numbers

We know that, sum of n natural numbers = ^{n (n + 1)}⁄₂

Thus, sum of first 10 multiples of 6 = 6 × ^{10 (10 + 1)}⁄₂

= 3 (10 × 11)

⇒ sum of first 10 multiples of 6 = 330

Now, we know that average

Thus, average of first 10 multiples of 6 = ³³⁰⁄₁₀ = 33

Thus, Average of first 10 multiples of 6 = 33 Answer

Alternate method

We have a List of the first 10 multiples of 6

= 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Thus, the sum of the first 10 multiples of 6

= 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60

= 330

Now, we know that average

Thus, average of first 10 multiples of 6 = ³³⁰⁄₁₀  = 33

Thus, Average of first 10 multiples of 6 = 33 Answer

(9) Find the average of first 20 multiples of 5

Solution :

Here, we have the List of the first 20 multiples of 5

= 5 (1, 2, 3, . . . , 20)

Now, sum of first 20 multiples of 5

= 5 (1 + 2 + 3 + . . . . + 20)

= 5 × ^{20 (20 + 1)}⁄₂

= 5 × ^{20 × 21}⁄₂

= 5 × 10 × 21

= 1050

Now, we know that, average

Therefore, average of first 20 multiples of 5 = ¹⁰⁵⁰⁄₂₀ = 52.5

⇒ average of first 20 multiples of 5 = 52.5 Answer

(10) Find the average of the first 50 multiples of 10.

Solution :

The first 50 multiples of 10

= 10 (1, 2, 3, . . . . . . , 50)

Here numbers in bracket are natural numbers

We know that, sum of n natural numbers = ^{n (n + 1)}⁄₂

Thus, sum of first 50 multiples of 10 = 10 × ^{50 (50 + 1)}⁄₂

= 5 × 50 × 51

Thus, sum of first 50 multiples of 10 = 12750

Now, we know that, average

Here, sum of given numbers = 12750

And, total numbers of given numbers = 50

Thus, Average = ¹²⁷⁵⁰⁄₅₀   = 255

Thus, Average of first 50 multiples of 10 = 255 Answer

(11) Calculate the average of first 500 multiples of 60.

Solution :

The first 500 multiples of 60

= 60 (1, 2, 3, . . . . . . , 500)

Here numbers in bracket are natural numbers

We know that, sum of n natural numbers = ^{n (n + 1)}⁄₂

Thus, sum of first 500 multiples of 60 = 60 × ^{500 (500 + 1)}⁄₂

= 30 × 500 × 501

Thus, sum of first 500 multiples of 60 = 7515000

Now, we know that, average

Here, sum of given numbers = 7515000

And, total numbers of given numbers = 500

Thus, Average = ^7515000⁄₅₀₀ = 15030

Thus, Average of first 500 multiples of 60 = 15030 Answer