Question : Find the average of even numbers from 12 to 366
Correct Answer 189
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 366
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 366 are
12, 14, 16, . . . . 366
After observing the above list of the even numbers from 12 to 366 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 366 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 366
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 366
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 even numbers from 12 to 366
= 12 + 366/2
= 378/2 = 189
Thus, the average of the even numbers from 12 to 366 = 189 Answer
Method (2) to find the average of the even numbers from 12 to 366
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 366 are
12, 14, 16, . . . . 366
The even numbers from 12 to 366 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 366
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 even numbers from 12 to 366
366 = 12 + (n – 1) × 2
⇒ 366 = 12 + 2 n – 2
⇒ 366 = 12 – 2 + 2 n
⇒ 366 = 10 + 2 n
After transposing 10 to LHS
⇒ 366 – 10 = 2 n
⇒ 356 = 2 n
After rearranging the above expression
⇒ 2 n = 356
After transposing 2 to RHS
⇒ n = 356/2
⇒ n = 178
Thus, the number of terms of even numbers from 12 to 366 = 178
This means 366 is the 178th term.
Finding the sum of the given even numbers from 12 to 366
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 even numbers from 12 to 366
= 178/2 (12 + 366)
= 178/2 × 378
= 178 × 378/2
= 67284/2 = 33642
Thus, the sum of all terms of the given even numbers from 12 to 366 = 33642
And, the total number of terms = 178
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 12 to 366
= 33642/178 = 189
Thus, the average of the given even numbers from 12 to 366 = 189 Answer
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