Question:
Find the average of even numbers from 12 to 298
Correct Answer
155
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 298
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 298 are
12, 14, 16, . . . . 298
After observing the above list of the even numbers from 12 to 298 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 298 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 298
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 298
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 298
= 12 + 298/2
= 310/2 = 155
Thus, the average of the even numbers from 12 to 298 = 155 Answer
Method (2) to find the average of the even numbers from 12 to 298
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 298 are
12, 14, 16, . . . . 298
The even numbers from 12 to 298 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 298
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 298
298 = 12 + (n – 1) × 2
⇒ 298 = 12 + 2 n – 2
⇒ 298 = 12 – 2 + 2 n
⇒ 298 = 10 + 2 n
After transposing 10 to LHS
⇒ 298 – 10 = 2 n
⇒ 288 = 2 n
After rearranging the above expression
⇒ 2 n = 288
After transposing 2 to RHS
⇒ n = 288/2
⇒ n = 144
Thus, the number of terms of even numbers from 12 to 298 = 144
This means 298 is the 144th term.
Finding the sum of the given even numbers from 12 to 298
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 298
= 144/2 (12 + 298)
= 144/2 × 310
= 144 × 310/2
= 44640/2 = 22320
Thus, the sum of all terms of the given even numbers from 12 to 298 = 22320
And, the total number of terms = 144
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 298
= 22320/144 = 155
Thus, the average of the given even numbers from 12 to 298 = 155 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 1958
(2) Find the average of the first 3050 even numbers.
(3) Find the average of even numbers from 12 to 1342
(4) Find the average of even numbers from 10 to 894
(5) Find the average of the first 3596 odd numbers.
(6) Find the average of odd numbers from 7 to 601
(7) Find the average of odd numbers from 13 to 317
(8) Find the average of even numbers from 4 to 1536
(9) Find the average of the first 3313 odd numbers.
(10) Find the average of the first 2338 even numbers.