Question : Find the average of even numbers from 12 to 246
Correct Answer 129
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 246
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 246 are
12, 14, 16, . . . . 246
After observing the above list of the even numbers from 12 to 246 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 246 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 246
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 246
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 246
= 12 + 246/2
= 258/2 = 129
Thus, the average of the even numbers from 12 to 246 = 129 Answer
Method (2) to find the average of the even numbers from 12 to 246
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 246 are
12, 14, 16, . . . . 246
The even numbers from 12 to 246 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 246
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 246
246 = 12 + (n – 1) × 2
⇒ 246 = 12 + 2 n – 2
⇒ 246 = 12 – 2 + 2 n
⇒ 246 = 10 + 2 n
After transposing 10 to LHS
⇒ 246 – 10 = 2 n
⇒ 236 = 2 n
After rearranging the above expression
⇒ 2 n = 236
After transposing 2 to RHS
⇒ n = 236/2
⇒ n = 118
Thus, the number of terms of even numbers from 12 to 246 = 118
This means 246 is the 118th term.
Finding the sum of the given even numbers from 12 to 246
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 246
= 118/2 (12 + 246)
= 118/2 × 258
= 118 × 258/2
= 30444/2 = 15222
Thus, the sum of all terms of the given even numbers from 12 to 246 = 15222
And, the total number of terms = 118
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 246
= 15222/118 = 129
Thus, the average of the given even numbers from 12 to 246 = 129 Answer
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