Average
MCQs Math


Question:     Find the average of odd numbers from 15 to 213


Correct Answer  114

Solution And Explanation

Solution

Method (1) to find the average of the odd numbers from 15 to 213

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

The odd numbers from 15 to 213 are

15, 17, 19, . . . . 213

After observing the above list of the odd numbers from 15 to 213 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 213 form an Arithmetic Series.

In the Arithmetic Series of the odd numbers from 15 to 213

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 213

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 213

= 15 + 213/2

= 228/2 = 114

Thus, the average of the odd numbers from 15 to 213 = 114 Answer

Method (2) to find the average of the odd numbers from 15 to 213

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

The odd numbers from 15 to 213 are

15, 17, 19, . . . . 213

The odd numbers from 15 to 213 form an Arithmetic Series in which

The First Term (a) = 15

The Common Difference (d) = 2

And the last term (ℓ) = 213

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 213

213 = 15 + (n – 1) × 2

⇒ 213 = 15 + 2 n – 2

⇒ 213 = 15 – 2 + 2 n

⇒ 213 = 13 + 2 n

After transposing 13 to LHS

⇒ 213 – 13 = 2 n

⇒ 200 = 2 n

After rearranging the above expression

⇒ 2 n = 200

After transposing 2 to RHS

⇒ n = 200/2

⇒ n = 100

Thus, the number of terms of odd numbers from 15 to 213 = 100

This means 213 is the 100th term.

Finding the sum of the given odd numbers from 15 to 213

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 213

= 100/2 (15 + 213)

= 100/2 × 228

= 100 × 228/2

= 22800/2 = 11400

Thus, the sum of all terms of the given odd numbers from 15 to 213 = 11400

And, the total number of terms = 100

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 213

= 11400/100 = 114

Thus, the average of the given odd numbers from 15 to 213 = 114 Answer


Similar Questions

(1) What will be the average of the first 4779 odd numbers?

(2) Find the average of odd numbers from 7 to 1125

(3) Find the average of odd numbers from 5 to 1489

(4) Find the average of the first 2066 even numbers.

(5) Find the average of odd numbers from 15 to 1293

(6) Find the average of the first 3230 odd numbers.

(7) Find the average of the first 2033 odd numbers.

(8) Find the average of the first 3923 odd numbers.

(9) Find the average of odd numbers from 9 to 377

(10) Find the average of odd numbers from 13 to 1379


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