Question:
Find the average of odd numbers from 15 to 543
Correct Answer
279
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 543
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 543 are
15, 17, 19, . . . . 543
After observing the above list of the odd numbers from 15 to 543 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 543 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 543
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 543
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 543
= 15 + 543/2
= 558/2 = 279
Thus, the average of the odd numbers from 15 to 543 = 279 Answer
Method (2) to find the average of the odd numbers from 15 to 543
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 543 are
15, 17, 19, . . . . 543
The odd numbers from 15 to 543 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 543
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 543
543 = 15 + (n – 1) × 2
⇒ 543 = 15 + 2 n – 2
⇒ 543 = 15 – 2 + 2 n
⇒ 543 = 13 + 2 n
After transposing 13 to LHS
⇒ 543 – 13 = 2 n
⇒ 530 = 2 n
After rearranging the above expression
⇒ 2 n = 530
After transposing 2 to RHS
⇒ n = 530/2
⇒ n = 265
Thus, the number of terms of odd numbers from 15 to 543 = 265
This means 543 is the 265th term.
Finding the sum of the given odd numbers from 15 to 543
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 543
= 265/2 (15 + 543)
= 265/2 × 558
= 265 × 558/2
= 147870/2 = 73935
Thus, the sum of all terms of the given odd numbers from 15 to 543 = 73935
And, the total number of terms = 265
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 543
= 73935/265 = 279
Thus, the average of the given odd numbers from 15 to 543 = 279 Answer
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