Average
MCQs Math


Question:     Find the average of even numbers from 6 to 940


Correct Answer  473

Solution And Explanation

Solution

Method (1) to find the average of the even numbers from 6 to 940

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

The even numbers from 6 to 940 are

6, 8, 10, . . . . 940

After observing the above list of the even numbers from 6 to 940 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 940 form an Arithmetic Series.

In the Arithmetic Series of the even numbers from 6 to 940

The First Term (a) = 6

The Common Difference (d) = 2

And the last term (ℓ) = 940

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 even numbers from 6 to 940

= 6 + 940/2

= 946/2 = 473

Thus, the average of the even numbers from 6 to 940 = 473 Answer

Method (2) to find the average of the even numbers from 6 to 940

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

The even numbers from 6 to 940 are

6, 8, 10, . . . . 940

The even numbers from 6 to 940 form an Arithmetic Series in which

The First Term (a) = 6

The Common Difference (d) = 2

And the last term (ℓ) = 940

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 even numbers from 6 to 940

940 = 6 + (n – 1) × 2

⇒ 940 = 6 + 2 n – 2

⇒ 940 = 6 – 2 + 2 n

⇒ 940 = 4 + 2 n

After transposing 4 to LHS

⇒ 940 – 4 = 2 n

⇒ 936 = 2 n

After rearranging the above expression

⇒ 2 n = 936

After transposing 2 to RHS

⇒ n = 936/2

⇒ n = 468

Thus, the number of terms of even numbers from 6 to 940 = 468

This means 940 is the 468th term.

Finding the sum of the given even numbers from 6 to 940

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 even numbers from 6 to 940

= 468/2 (6 + 940)

= 468/2 × 946

= 468 × 946/2

= 442728/2 = 221364

Thus, the sum of all terms of the given even numbers from 6 to 940 = 221364

And, the total number of terms = 468

Since, the average of the given numbers

= Sum of the given numbers/Total number of given numbers

Thus, the average of the given even numbers from 6 to 940

= 221364/468 = 473

Thus, the average of the given even numbers from 6 to 940 = 473 Answer


Similar Questions

(1) Find the average of the first 4487 even numbers.

(2) Find the average of even numbers from 8 to 202

(3) Find the average of the first 1227 odd numbers.

(4) Find the average of odd numbers from 9 to 251

(5) What is the average of the first 783 even numbers?

(6) Find the average of the first 4931 even numbers.

(7) Find the average of even numbers from 4 to 1894

(8) Find the average of the first 3216 even numbers.

(9) Find the average of odd numbers from 13 to 737

(10) Find the average of the first 892 odd numbers.


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