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MCQs Math


Question:     Find the average of odd numbers from 15 to 479


Correct Answer  247

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 479

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

The odd numbers from 15 to 479 are

15, 17, 19, . . . . 479

After observing the above list of the odd numbers from 15 to 479 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 479 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 479

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 479

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 479

= 15 + 479/2

= 494/2 = 247

Thus, the average of the odd numbers from 15 to 479 = 247 Answer

Method (2) to find the average of the odd numbers from 15 to 479

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

The odd numbers from 15 to 479 are

15, 17, 19, . . . . 479

The odd numbers from 15 to 479 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 479

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 479

479 = 15 + (n – 1) × 2

⇒ 479 = 15 + 2 n – 2

⇒ 479 = 15 – 2 + 2 n

⇒ 479 = 13 + 2 n

After transposing 13 to LHS

⇒ 479 – 13 = 2 n

⇒ 466 = 2 n

After rearranging the above expression

⇒ 2 n = 466

After transposing 2 to RHS

⇒ n = 466/2

⇒ n = 233

Thus, the number of terms of odd numbers from 15 to 479 = 233

This means 479 is the 233th term.

Finding the sum of the given odd numbers from 15 to 479

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 479

= 233/2 (15 + 479)

= 233/2 × 494

= 233 × 494/2

= 115102/2 = 57551

Thus, the sum of all terms of the given odd numbers from 15 to 479 = 57551

And, the total number of terms = 233

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 479

= 57551/233 = 247

Thus, the average of the given odd numbers from 15 to 479 = 247 Answer


Similar Questions

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(3) What is the average of the first 104 even numbers?

(4) Find the average of the first 2527 odd numbers.

(5) Find the average of odd numbers from 11 to 1315

(6) Find the average of even numbers from 10 to 1166

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