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Math MCQs


Question :    Find the average of odd numbers from 15 to 903


Correct Answer  459

Solution & Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 903

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

The odd numbers from 15 to 903 are

15, 17, 19, . . . . 903

After observing the above list of the odd numbers from 15 to 903 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 903 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 903

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 903

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 903

= 15 + 903/2

= 918/2 = 459

Thus, the average of the odd numbers from 15 to 903 = 459 Answer

Method (2) to find the average of the odd numbers from 15 to 903

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

The odd numbers from 15 to 903 are

15, 17, 19, . . . . 903

The odd numbers from 15 to 903 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 903

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 903

903 = 15 + (n – 1) × 2

⇒ 903 = 15 + 2 n – 2

⇒ 903 = 15 – 2 + 2 n

⇒ 903 = 13 + 2 n

After transposing 13 to LHS

⇒ 903 – 13 = 2 n

⇒ 890 = 2 n

After rearranging the above expression

⇒ 2 n = 890

After transposing 2 to RHS

⇒ n = 890/2

⇒ n = 445

Thus, the number of terms of odd numbers from 15 to 903 = 445

This means 903 is the 445th term.

Finding the sum of the given odd numbers from 15 to 903

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 903

= 445/2 (15 + 903)

= 445/2 × 918

= 445 × 918/2

= 408510/2 = 204255

Thus, the sum of all terms of the given odd numbers from 15 to 903 = 204255

And, the total number of terms = 445

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 903

= 204255/445 = 459

Thus, the average of the given odd numbers from 15 to 903 = 459 Answer


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