Question : Find the average of even numbers from 12 to 644
Correct Answer 328
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 644
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 644 are
12, 14, 16, . . . . 644
After observing the above list of the even numbers from 12 to 644 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 644 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 644
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 644
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 644
= 12 + 644/2
= 656/2 = 328
Thus, the average of the even numbers from 12 to 644 = 328 Answer
Method (2) to find the average of the even numbers from 12 to 644
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 644 are
12, 14, 16, . . . . 644
The even numbers from 12 to 644 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 644
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 644
644 = 12 + (n – 1) × 2
⇒ 644 = 12 + 2 n – 2
⇒ 644 = 12 – 2 + 2 n
⇒ 644 = 10 + 2 n
After transposing 10 to LHS
⇒ 644 – 10 = 2 n
⇒ 634 = 2 n
After rearranging the above expression
⇒ 2 n = 634
After transposing 2 to RHS
⇒ n = 634/2
⇒ n = 317
Thus, the number of terms of even numbers from 12 to 644 = 317
This means 644 is the 317th term.
Finding the sum of the given even numbers from 12 to 644
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 644
= 317/2 (12 + 644)
= 317/2 × 656
= 317 × 656/2
= 207952/2 = 103976
Thus, the sum of all terms of the given even numbers from 12 to 644 = 103976
And, the total number of terms = 317
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 644
= 103976/317 = 328
Thus, the average of the given even numbers from 12 to 644 = 328 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1120
(2) Find the average of the first 3816 odd numbers.
(3) Find the average of the first 952 odd numbers.
(4) Find the average of odd numbers from 3 to 1387
(5) Find the average of odd numbers from 5 to 755
(6) Find the average of odd numbers from 13 to 873
(7) Find the average of odd numbers from 7 to 1353
(8) Find the average of the first 3479 even numbers.
(9) Find the average of the first 3280 even numbers.
(10) Find the average of even numbers from 6 to 954