Average
MCQs Math


Question:     Find the average of odd numbers from 9 to 329


Correct Answer  169

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 9 to 329

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

The odd numbers from 9 to 329 are

9, 11, 13, . . . . 329

After observing the above list of the odd numbers from 9 to 329 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 329 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 9 to 329

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 329

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 9 to 329

= 9 + 329/2

= 338/2 = 169

Thus, the average of the odd numbers from 9 to 329 = 169 Answer

Method (2) to find the average of the odd numbers from 9 to 329

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

The odd numbers from 9 to 329 are

9, 11, 13, . . . . 329

The odd numbers from 9 to 329 form an Arithmetic Series in which

The First Term (a) = 9

The Common Difference (d) = 2

And the last term (ℓ) = 329

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 9 to 329

329 = 9 + (n – 1) × 2

⇒ 329 = 9 + 2 n – 2

⇒ 329 = 9 – 2 + 2 n

⇒ 329 = 7 + 2 n

After transposing 7 to LHS

⇒ 329 – 7 = 2 n

⇒ 322 = 2 n

After rearranging the above expression

⇒ 2 n = 322

After transposing 2 to RHS

⇒ n = 322/2

⇒ n = 161

Thus, the number of terms of odd numbers from 9 to 329 = 161

This means 329 is the 161th term.

Finding the sum of the given odd numbers from 9 to 329

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 9 to 329

= 161/2 (9 + 329)

= 161/2 × 338

= 161 × 338/2

= 54418/2 = 27209

Thus, the sum of all terms of the given odd numbers from 9 to 329 = 27209

And, the total number of terms = 161

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 9 to 329

= 27209/161 = 169

Thus, the average of the given odd numbers from 9 to 329 = 169 Answer


Similar Questions

(1) What is the average of the first 1573 even numbers?

(2) Find the average of the first 2983 even numbers.

(3) What is the average of the first 1040 even numbers?

(4) Find the average of odd numbers from 15 to 529

(5) Find the average of even numbers from 12 to 312

(6) Find the average of odd numbers from 13 to 875

(7) Find the average of even numbers from 10 to 1780

(8) Find the average of odd numbers from 11 to 489

(9) What is the average of the first 1243 even numbers?

(10) Find the average of the first 2493 even numbers.


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