Question : Find the average of even numbers from 12 to 848
Correct Answer 430
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 848
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 848 are
12, 14, 16, . . . . 848
After observing the above list of the even numbers from 12 to 848 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 848 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 848
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 848
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 848
= 12 + 848/2
= 860/2 = 430
Thus, the average of the even numbers from 12 to 848 = 430 Answer
Method (2) to find the average of the even numbers from 12 to 848
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 848 are
12, 14, 16, . . . . 848
The even numbers from 12 to 848 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 848
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 848
848 = 12 + (n – 1) × 2
⇒ 848 = 12 + 2 n – 2
⇒ 848 = 12 – 2 + 2 n
⇒ 848 = 10 + 2 n
After transposing 10 to LHS
⇒ 848 – 10 = 2 n
⇒ 838 = 2 n
After rearranging the above expression
⇒ 2 n = 838
After transposing 2 to RHS
⇒ n = 838/2
⇒ n = 419
Thus, the number of terms of even numbers from 12 to 848 = 419
This means 848 is the 419th term.
Finding the sum of the given even numbers from 12 to 848
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 848
= 419/2 (12 + 848)
= 419/2 × 860
= 419 × 860/2
= 360340/2 = 180170
Thus, the sum of all terms of the given even numbers from 12 to 848 = 180170
And, the total number of terms = 419
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 848
= 180170/419 = 430
Thus, the average of the given even numbers from 12 to 848 = 430 Answer
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