Question : Find the average of odd numbers from 9 to 893
Correct Answer 451
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 9 to 893
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 9 to 893 are
9, 11, 13, . . . . 893
After observing the above list of the odd numbers from 9 to 893 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 9 to 893 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 9 to 893
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 893
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 9 to 893
= 9 + 893/2
= 902/2 = 451
Thus, the average of the odd numbers from 9 to 893 = 451 Answer
Method (2) to find the average of the odd numbers from 9 to 893
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 9 to 893 are
9, 11, 13, . . . . 893
The odd numbers from 9 to 893 form an Arithmetic Series in which
The First Term (a) = 9
The Common Difference (d) = 2
And the last term (ℓ) = 893
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 9 to 893
893 = 9 + (n – 1) × 2
⇒ 893 = 9 + 2 n – 2
⇒ 893 = 9 – 2 + 2 n
⇒ 893 = 7 + 2 n
After transposing 7 to LHS
⇒ 893 – 7 = 2 n
⇒ 886 = 2 n
After rearranging the above expression
⇒ 2 n = 886
After transposing 2 to RHS
⇒ n = 886/2
⇒ n = 443
Thus, the number of terms of odd numbers from 9 to 893 = 443
This means 893 is the 443th term.
Finding the sum of the given odd numbers from 9 to 893
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 9 to 893
= 443/2 (9 + 893)
= 443/2 × 902
= 443 × 902/2
= 399586/2 = 199793
Thus, the sum of all terms of the given odd numbers from 9 to 893 = 199793
And, the total number of terms = 443
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 9 to 893
= 199793/443 = 451
Thus, the average of the given odd numbers from 9 to 893 = 451 Answer
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