Question:
Find the average of odd numbers from 11 to 105
Correct Answer
58
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 105
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 105 are
11, 13, 15, . . . . 105
After observing the above list of the odd numbers from 11 to 105 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 105 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 105
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 105
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 11 to 105
= 11 + 105/2
= 116/2 = 58
Thus, the average of the odd numbers from 11 to 105 = 58 Answer
Method (2) to find the average of the odd numbers from 11 to 105
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 105 are
11, 13, 15, . . . . 105
The odd numbers from 11 to 105 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 105
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 11 to 105
105 = 11 + (n – 1) × 2
⇒ 105 = 11 + 2 n – 2
⇒ 105 = 11 – 2 + 2 n
⇒ 105 = 9 + 2 n
After transposing 9 to LHS
⇒ 105 – 9 = 2 n
⇒ 96 = 2 n
After rearranging the above expression
⇒ 2 n = 96
After transposing 2 to RHS
⇒ n = 96/2
⇒ n = 48
Thus, the number of terms of odd numbers from 11 to 105 = 48
This means 105 is the 48th term.
Finding the sum of the given odd numbers from 11 to 105
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 11 to 105
= 48/2 (11 + 105)
= 48/2 × 116
= 48 × 116/2
= 5568/2 = 2784
Thus, the sum of all terms of the given odd numbers from 11 to 105 = 2784
And, the total number of terms = 48
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 11 to 105
= 2784/48 = 58
Thus, the average of the given odd numbers from 11 to 105 = 58 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1407
(2) Find the average of odd numbers from 15 to 233
(3) What is the average of the first 1488 even numbers?
(4) Find the average of the first 2168 even numbers.
(5) Find the average of odd numbers from 11 to 53
(6) Find the average of the first 3267 even numbers.
(7) Find the average of the first 2183 odd numbers.
(8) Find the average of odd numbers from 13 to 723
(9) Find the average of the first 2687 even numbers.
(10) Find the average of the first 4220 even numbers.