Question : Find the average of odd numbers from 11 to 885
Correct Answer 448
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 885
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 885 are
11, 13, 15, . . . . 885
After observing the above list of the odd numbers from 11 to 885 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 885 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 885
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 885
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 885
= 11 + 885/2
= 896/2 = 448
Thus, the average of the odd numbers from 11 to 885 = 448 Answer
Method (2) to find the average of the odd numbers from 11 to 885
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 885 are
11, 13, 15, . . . . 885
The odd numbers from 11 to 885 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 885
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 885
885 = 11 + (n – 1) × 2
⇒ 885 = 11 + 2 n – 2
⇒ 885 = 11 – 2 + 2 n
⇒ 885 = 9 + 2 n
After transposing 9 to LHS
⇒ 885 – 9 = 2 n
⇒ 876 = 2 n
After rearranging the above expression
⇒ 2 n = 876
After transposing 2 to RHS
⇒ n = 876/2
⇒ n = 438
Thus, the number of terms of odd numbers from 11 to 885 = 438
This means 885 is the 438th term.
Finding the sum of the given odd numbers from 11 to 885
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 885
= 438/2 (11 + 885)
= 438/2 × 896
= 438 × 896/2
= 392448/2 = 196224
Thus, the sum of all terms of the given odd numbers from 11 to 885 = 196224
And, the total number of terms = 438
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 885
= 196224/438 = 448
Thus, the average of the given odd numbers from 11 to 885 = 448 Answer
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