Question : Find the average of odd numbers from 15 to 405
Correct Answer 210
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 405
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 405 are
15, 17, 19, . . . . 405
After observing the above list of the odd numbers from 15 to 405 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 405 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 405
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 405
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 405
= 15 + 405/2
= 420/2 = 210
Thus, the average of the odd numbers from 15 to 405 = 210 Answer
Method (2) to find the average of the odd numbers from 15 to 405
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 405 are
15, 17, 19, . . . . 405
The odd numbers from 15 to 405 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 405
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 405
405 = 15 + (n – 1) × 2
⇒ 405 = 15 + 2 n – 2
⇒ 405 = 15 – 2 + 2 n
⇒ 405 = 13 + 2 n
After transposing 13 to LHS
⇒ 405 – 13 = 2 n
⇒ 392 = 2 n
After rearranging the above expression
⇒ 2 n = 392
After transposing 2 to RHS
⇒ n = 392/2
⇒ n = 196
Thus, the number of terms of odd numbers from 15 to 405 = 196
This means 405 is the 196th term.
Finding the sum of the given odd numbers from 15 to 405
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 405
= 196/2 (15 + 405)
= 196/2 × 420
= 196 × 420/2
= 82320/2 = 41160
Thus, the sum of all terms of the given odd numbers from 15 to 405 = 41160
And, the total number of terms = 196
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 405
= 41160/196 = 210
Thus, the average of the given odd numbers from 15 to 405 = 210 Answer
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