Question:
Find the average of even numbers from 12 to 894
Correct Answer
453
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 894
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 894 are
12, 14, 16, . . . . 894
After observing the above list of the even numbers from 12 to 894 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 894 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 894
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 894
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 894
= 12 + 894/2
= 906/2 = 453
Thus, the average of the even numbers from 12 to 894 = 453 Answer
Method (2) to find the average of the even numbers from 12 to 894
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 894 are
12, 14, 16, . . . . 894
The even numbers from 12 to 894 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 894
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 894
894 = 12 + (n – 1) × 2
⇒ 894 = 12 + 2 n – 2
⇒ 894 = 12 – 2 + 2 n
⇒ 894 = 10 + 2 n
After transposing 10 to LHS
⇒ 894 – 10 = 2 n
⇒ 884 = 2 n
After rearranging the above expression
⇒ 2 n = 884
After transposing 2 to RHS
⇒ n = 884/2
⇒ n = 442
Thus, the number of terms of even numbers from 12 to 894 = 442
This means 894 is the 442th term.
Finding the sum of the given even numbers from 12 to 894
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 894
= 442/2 (12 + 894)
= 442/2 × 906
= 442 × 906/2
= 400452/2 = 200226
Thus, the sum of all terms of the given even numbers from 12 to 894 = 200226
And, the total number of terms = 442
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 894
= 200226/442 = 453
Thus, the average of the given even numbers from 12 to 894 = 453 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 229
(2) Find the average of the first 1514 odd numbers.
(3) Find the average of odd numbers from 5 to 955
(4) Find the average of even numbers from 12 to 1682
(5) Find the average of the first 2293 even numbers.
(6) Find the average of odd numbers from 13 to 469
(7) Find the average of the first 3714 odd numbers.
(8) Find the average of the first 293 odd numbers.
(9) Find the average of odd numbers from 15 to 1423
(10) Find the average of odd numbers from 5 to 899