10upon10.com

Average
Math MCQs


Question :    Find the average of odd numbers from 15 to 593


Correct Answer  304

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 593

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

The odd numbers from 15 to 593 are

15, 17, 19, . . . . 593

After observing the above list of the odd numbers from 15 to 593 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 593 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 593

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 593

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 15 to 593

= 15 + 593/2

= 608/2 = 304

Thus, the average of the odd numbers from 15 to 593 = 304 Answer

Method (2) to find the average of the odd numbers from 15 to 593

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

The odd numbers from 15 to 593 are

15, 17, 19, . . . . 593

The odd numbers from 15 to 593 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 593

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 15 to 593

593 = 15 + (n – 1) × 2

⇒ 593 = 15 + 2 n – 2

⇒ 593 = 15 – 2 + 2 n

⇒ 593 = 13 + 2 n

After transposing 13 to LHS

⇒ 593 – 13 = 2 n

⇒ 580 = 2 n

After rearranging the above expression

⇒ 2 n = 580

After transposing 2 to RHS

⇒ n = 580/2

⇒ n = 290

Thus, the number of terms of odd numbers from 15 to 593 = 290

This means 593 is the 290th term.

Finding the sum of the given odd numbers from 15 to 593

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 15 to 593

= 290/2 (15 + 593)

= 290/2 × 608

= 290 × 608/2

= 176320/2 = 88160

Thus, the sum of all terms of the given odd numbers from 15 to 593 = 88160

And, the total number of terms = 290

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 15 to 593

= 88160/290 = 304

Thus, the average of the given odd numbers from 15 to 593 = 304 Answer


Similar Questions

(1) Find the average of the first 263 odd numbers.

(2) Find the average of odd numbers from 11 to 919

(3) Find the average of the first 2577 even numbers.

(4) Find the average of the first 2393 even numbers.

(5) What will be the average of the first 4332 odd numbers?

(6) What will be the average of the first 4326 odd numbers?

(7) Find the average of odd numbers from 13 to 895

(8) Find the average of even numbers from 8 to 1044

(9) Find the average of odd numbers from 13 to 503

(10) Find the average of odd numbers from 5 to 127