Average
MCQs Math


Question:     Find the average of odd numbers from 11 to 201


Correct Answer  106

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 11 to 201

Shortcut Trick to find the average of the given continuous odd numbers

The odd numbers from 11 to 201 are

11, 13, 15, . . . . 201

After observing the above list of the odd numbers from 11 to 201 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 201 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 11 to 201

The First Term (a) = 11

The Common Difference (d) = 2

And the last term (ℓ) = 201

The average of the numbers forming an Arithmetic Series

= The first term (a) + The last term (ℓ)/2

⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2

Thus, the average of the odd numbers from 11 to 201

= 11 + 201/2

= 212/2 = 106

Thus, the average of the odd numbers from 11 to 201 = 106 Answer

Method (2) to find the average of the odd numbers from 11 to 201

Finding the average of given continuous odd numbers after finding their sum

The odd numbers from 11 to 201 are

11, 13, 15, . . . . 201

The odd numbers from 11 to 201 form an Arithmetic Series in which

The First Term (a) = 11

The Common Difference (d) = 2

And the last term (ℓ) = 201

The Average of the given numbers

= Sum of the given numbers/Total number of given numbers

Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers

Finding the number of terms

For an Arithmetic Series, the nth term

an = a + (n – 1) d

Where

a = First term

d = Common difference

n = number of terms

an = nth term

Thus, for the given series of the odd numbers from 11 to 201

201 = 11 + (n – 1) × 2

⇒ 201 = 11 + 2 n – 2

⇒ 201 = 11 – 2 + 2 n

⇒ 201 = 9 + 2 n

After transposing 9 to LHS

⇒ 201 – 9 = 2 n

⇒ 192 = 2 n

After rearranging the above expression

⇒ 2 n = 192

After transposing 2 to RHS

⇒ n = 192/2

⇒ n = 96

Thus, the number of terms of odd numbers from 11 to 201 = 96

This means 201 is the 96th term.

Finding the sum of the given odd numbers from 11 to 201

The sum of all terms (S) in an Arithmetic Series

= n/2 (a + ℓ)

Where, n = number of terms

a = First term

And, ℓ = Last term

Thus, the sum of all terms (S) of the given odd numbers from 11 to 201

= 96/2 (11 + 201)

= 96/2 × 212

= 96 × 212/2

= 20352/2 = 10176

Thus, the sum of all terms of the given odd numbers from 11 to 201 = 10176

And, the total number of terms = 96

Since, the average of the given numbers

= Sum of the given numbers/Total number of given numbers

Thus, the average of the given odd numbers from 11 to 201

= 10176/96 = 106

Thus, the average of the given odd numbers from 11 to 201 = 106 Answer


Similar Questions

(1) Find the average of the first 1614 odd numbers.

(2) Find the average of odd numbers from 5 to 1477

(3) Find the average of the first 1003 odd numbers.

(4) What is the average of the first 1995 even numbers?

(5) Find the average of the first 1812 odd numbers.

(6) What will be the average of the first 4922 odd numbers?

(7) What is the average of the first 77 odd numbers?

(8) Find the average of the first 2187 odd numbers.

(9) Find the average of even numbers from 10 to 838

(10) Find the average of odd numbers from 13 to 99


NCERT Solution and CBSE Notes for class twelve, eleventh, tenth, ninth, seventh, sixth, fifth, fourth and General Math for competitive Exams. ©