Average
MCQs Math


Question:     Find the average of odd numbers from 3 to 359


Correct Answer  181

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 3 to 359

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

The odd numbers from 3 to 359 are

3, 5, 7, . . . . 359

After observing the above list of the odd numbers from 3 to 359 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 359 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 3 to 359

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 359

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 3 to 359

= 3 + 359/2

= 362/2 = 181

Thus, the average of the odd numbers from 3 to 359 = 181 Answer

Method (2) to find the average of the odd numbers from 3 to 359

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

The odd numbers from 3 to 359 are

3, 5, 7, . . . . 359

The odd numbers from 3 to 359 form an Arithmetic Series in which

The First Term (a) = 3

The Common Difference (d) = 2

And the last term (ℓ) = 359

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 3 to 359

359 = 3 + (n – 1) × 2

⇒ 359 = 3 + 2 n – 2

⇒ 359 = 3 – 2 + 2 n

⇒ 359 = 1 + 2 n

After transposing 1 to LHS

⇒ 359 – 1 = 2 n

⇒ 358 = 2 n

After rearranging the above expression

⇒ 2 n = 358

After transposing 2 to RHS

⇒ n = 358/2

⇒ n = 179

Thus, the number of terms of odd numbers from 3 to 359 = 179

This means 359 is the 179th term.

Finding the sum of the given odd numbers from 3 to 359

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 3 to 359

= 179/2 (3 + 359)

= 179/2 × 362

= 179 × 362/2

= 64798/2 = 32399

Thus, the sum of all terms of the given odd numbers from 3 to 359 = 32399

And, the total number of terms = 179

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 3 to 359

= 32399/179 = 181

Thus, the average of the given odd numbers from 3 to 359 = 181 Answer


Similar Questions

(1) Find the average of even numbers from 4 to 1324

(2) Find the average of even numbers from 12 to 590

(3) Find the average of odd numbers from 7 to 935

(4) Find the average of the first 2832 odd numbers.

(5) Find the average of the first 4163 even numbers.

(6) Find the average of the first 2362 even numbers.

(7) Find the average of the first 2826 even numbers.

(8) Find the average of the first 3284 odd numbers.

(9) Find the average of even numbers from 12 to 1988

(10) Find the average of the first 3548 even numbers.


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