Average - Arithmetic

(1) If average of 3 consecutive natural numbers is 3 then find the numbers.

Solution:

Given, the average of 3 consecutive natural numbers = 3

Therefore, numbers = ?

Let first number = a

Therefore, second consecutive number = a + 1

And, third consecutive number = a + 1 + 1 = a + 2

Now, we know that, Average given observations

Thus, average of given numbers

After cross multiplication, we get

3 × 3 = 3a + 3

⇒ 9 = 3a + 3

⇒ 3a + 3 = 9

⇒ 3a = 9 – 3

⇒ 3a = 6

Thus, second number = a + 1 = 2 + 1 = 3

And third number = a + 2 = 2 + 2 = 4

Thus, numbers are 2, 3 and 4 Answer

Shortcut Method/Trick to solve the question

Average means "Mean". This means lie in middle.

Here given, average of three consecutive numbers = 3

This means the number 3 lies in the middle among three required numbers.

Thus, if 3 is in the middle

Then, First number = Number in middle – 1

= 3 – 1 = 2

And, third number = Number in middle + 1

= 3 + 1 = 4

Thus, numbers are 2, 3, and 4 Answer

(2) If average of three consecutive numbers is 6 then find the smallest number.

Solution :

Given, average of three consecutive numbers = 6

Then, smallest number = ?

Let, first number = a

Therefore, second number = a + 1

And third number = a + 1 + 1 = a + 2

Now, we know that Average of given observations

Thus, average of given numbers

After cross multiplication, we get

6 × 3 = 3a + 3

⇒ 18 = 3a + 3

⇒ 3a + 3 = 18

⇒ 3a = 18 – 3

⇒ 3a = 15

Thus, a = 15/3 =5

Thus, smallest number = 5 Answer

Shortcut Method/Trick to solve the question

Average means "Mean". This means lie in middle.

Here given, average of three consecutive numbers = 6

This means the number 6 lies in the middle among three required numbers.

Thus, if 3 is in the middle

Then, First number = Number in middle – 1

= 6 – 1 = 5

And, third number = Number in middle + 1

= 6 + 1 = 7

Thus, numbers are 5, 6, and 7

Clearly, smallest number = 5 Answer

(3) If the average of 4 consecutive numbers is 4.5 then find the largest number.

Solution

Given, average of 4 consecutive numbers = 4.5

Thus, largest number = ?

Let, first number = a

Therefore, second number = a + 1

And third number = second number + 1

= a + 1 + 1 = a + 2

And, fourth number = third number + 1

= a + 2 + 1 = a + 3

Now, we know that, Average of given observations

Thus, average of given numbers

After cross multiplication, we get

4.5 × 4 = 4a + 6

⇒ 18 = 4a + 6

⇒ 4a + 6 = 18

⇒ 4a = 18 – 6

⇒ 4a = 12

Now, largest number = a + 3

= 3 + 3 = 6

Thus, largest number among the given consecutive numbers = 6 Answer

Shortcut / trick to solve the question

Average means "Mean". This means lie in middle.

Here, given average of 4 consecutive numbers = 4.5

This means average lie between 2^nd and 3^rd numbers.

Since average is 4.5, it means 2^nd number is 4 and 3^rd number is 5.

Clearly, four consecutive numbers are 3, 4, 5 and 6

And hence largest among them is 6 Answer.

(4) The average of two consecutive odd numbers is 6, then find the numbers.

Solution :

Given, Average of two consecutive odd numbers = 6

Then numbers = ?

Let first odd number = a

Thus, second odd number = a + 2

Now, we know that, Average of given observations

⇒ 6

After cross multiplication, we get

⇒ 6 × 2 = 2a + 2

⇒ 12 = 2a + 2

⇒ 2a + 2 = 12

⇒ 2a = 12 – 2

⇒ 2a = 10

Thus, a + 2 = 5 + 2 = 7

Therefore, numbers are 5 and 7 Answer

Shortcut/Trick to solve the question

Average means "Mean". This means lie in middle.

Here, given, average of two odd number = 6

This means 6 is between two given numbers which are odd.

Thus, first number is 6 – 1 = 5

and second number is 6 + 1 = 7

Thus, required odd numbers are 5 and 7 Answer

(5) Average of three consecutive odd numbers is 11 then find the numbers.

Solution :

Given, average of three consecutive odd numbers = 11

Thus, numbers = ?

Let first number = a

Then, second odd number = a + 2

And third odd numbers = a + 2 + 2 = a + 4

Now, we know that, Average of given observations

⇒ 11

⇒ 11

After cross multiplication, we get

11 × 3 = 3a + 6

⇒ 33 = 3a + 6

⇒ 3a + 6 = 33

⇒ 3a = 33 – 6

⇒ 3a = 27

Thus,

Thus, a + 2 = 9 + 2 = 11

And a + 4 = 9 + 4 = 13

Thus, numbers area 9, 11 and 13 Answer

Shortcut method/Trick to solve the question

We know that, Average means "Mean". This means lie in middle.

Here given 11 is the average of three odd numbers.

This means 11 lies in the middle among the three odd numbers.

Means 11 is the second odd number.

Thus, first odd number = 11 – 2 = 9

And third odd number = 11 + 2 = 13

Thus, required odd numbers are 9, 11 and 13 Answer

(6) Average of four consecutive odd numbers is 16 then find the numbers.

Solution

Given, average of four consecutive odd numbers = 16

Thus, numbers = ?

Let first odd number = a

Then second odd number = a + 2

And, third odd number = a + 2 + 2 = a + 4

And fourth odd number = a + 4 + 2 = a + 6

Now, we know that, Average of given observations

Thus, average of given numbers

⇒ 16

After cross multiplication we get

⇒ 16 × 4 = 4 a + 12

⇒ 64 = 4 a + 12

⇒ 4 a + 12 = 64

⇒ 4 a = 64 – 12

⇒ 4 a = 52

Therefore,

Thus, a + 2 = 13 + 2 = 15

And a + 4 = 13 + 4 = 17

And a + 6 = 13 + 6 = 19

Thus, numbers area 13, 15, 17 and 19 Answer

Shortcut method/Trick to solve the question

We know that, Average means "Mean". This means lie in middle.

Here given, 16 is the average of four consecutive odd numbers.

This means 16 lies between 2^nd and 3^rd numbers.

Since, numbers are odd, this means difference between 2 consecutive numbers = 2

Thus, 2^nd number = 16 – 1 = 15

And 3^rd number = 16 + 1 = 17

Thus, 1^st number = 15 – 2 = 13

And 4^th number = 17 + 2 = 19

Thus, numbers are 13, 15, 17 and 19 Answer

(7) Average of three consecutive even numbers is 20, then find the numbers.

Solution :

Given, average of three consecutive even number = 20

Then, numbers = ?

Let the first number = a

Then second even number = a + 2

And third even number = a + 2 + 2 = a + 4

Now, we know that, Average of given observations

Thus, Average of given numbers

⇒ 20

After cross multiplication, we get

⇒ 20 × 3 = 3a + 6

⇒ 60 = 3a + 6

⇒ 3a + 6 = 60

⇒ 3a = 60 – 6

⇒ 3a = 54

Therefore, a = 54/3 = 18

Thus, a + 2 = 18 + 2 = 20

And, a + 4 = 18 + 4 = 22

Thus, numbers are 18, 20 and 22 Answer

Shortcut method/Trick to solve the question

We know that, Average means "Mean". This means lie in middle.

Here given, 20 is the average of three even number. This means 20 is in the middle of the three even numbers.

Thus, 1^st number = 20 – 2 = 18

And 3^rd number = 20 + 2 = 22

Thus, numbers are 18, 20 and 22 Answer

(8) Average of 7 consecutive even numbers is 50. Then find the largest number.

Solution :

Given, average of 7 consecutive numbers = 50

Then largest among them = ?

Let, first number = a

Then, second number = a + 2

And, third number = a + 2 + 2 = a + 4

And, fourth number = third number + 2

= a + 4 + 2 = a + 6

And, fifth number = fourth number + 2

= a + 6 + 2 = a + 8

And, sixth number = fifth number + 2

= a + 8 + 2 = a + 10

And seventh number = sixth number + 2

= a + 10 + 2 = a + 12

Sum of given numbers = a + a + 2 + a + 4 + a + 6 + a + 8 + a + 10 + a + 12

= 7a + 42

Now, we know that, Average of given observations

⇒ Average of given numbers

⇒ 50

After cross multiplication, we get

⇒ 50 × 7 = 7a + 42

⇒ 350 = 7a + 42

⇒ 7a + 42 = 350

⇒ 7a = 350 – 42

⇒ 7a = 308

Thus, second number = a + 2

= 44 + 2 = 46

And, third number = a + 4

= 44 + 4 = 48

And fourth number = a + 6

= 44 + 6 = 50

And fifth number = a + 8

= 44 + 8 = 52

And sixth number = a + 10

= 44 + 10 = 54

And, seventh number = a + 12

= 44 + 12 = 56

Thus, numbers are 44, 46, 48, 50, 52, 54 and 56

Thus, 56 is the largest number.

Shortcut method/Trick to solve the question

We know that, Average means "Mean". This means lie in middle.

Here given, 50 is the average of 7 consecutive even numbers.

4 lies in the middle of 7.

Thus, 50 is the 4^th number.

Thus, 7^th number is the largest.

Since, 50 is the 4^th number

Thus, 50 + 2 = 52 is the 5^th number

And, 52 + 2 = 54 is the 6^th number

And, 54 + 2 = 56 is the 7^th number

Thus, largest number = 56 Answer

 

(9) If the average of 5 consecutive natural numbers is 7, the find the smallest and largest of these number.

Solution :

Given, average of 5 consecutive natural numbers = 7

Thus, smallest and largest number = ?

Let first number = a

Therefore, second consecutive natural number = a + 1

And, third consecutive number = a + 1 + 1 = a + 2

And, fourth consecutive number = a + 2 + 1 = a + 3

And, fifth number = a + 3 + 1 = a + 4

Now, sum of given 5 numbers = a + (a + 1) + (a + 2) + (a + 3) + (a + 4)

= a + a + 1 + a + 2 + a + 3 + a + 4

= 5a + 10

Now, we know that, Average given observations

Thus, Average of given 5 consecutive numbers

After cross multiplication, we get

⇒ 7 × 5 = 5 a + 10

⇒ 35 = 5 a + 10

⇒ 5 a + 10 = 35

⇒ 5 a = 35 – 10

⇒ 5 a = 25

Thus, smallest number = 5

And largest number = a + 4 = 5 + 4 =9

Thus, smallest number = 5 and largest number = 9 Answer

Shortcut method/Trick to solve the question

We know that, Average means "Mean". This means lie in middle.

Here given, 7 is the average of 5 consecutive numbers.

This means 7 lies on the 3^rd number.

This means 7 is the third number.

And, 2^nd number = 7 – 1 = 6

And, 1^st number = 7 – 2 = 5

And, 4^th number = 7 + 1 = 8

And, 5^th number = 7 + 2 = 9

Thus, smallest number = 5 and largest number = 9 Answer

 

(10) Find the average of first 5 prime numbers.

Solution :

Numbers greater than 1 having no factor other than number itself and 1 are called Prime Numbers.

List of first 5 prime numbers is

3, 5, 7, 11, and 13

We know that, average

Thus, Average of given numbers

=39/5 =7.8

Thus, Average of first 5 prime numbers = 7.8 Answer