Question:
Find the average of even numbers from 12 to 338
Correct Answer
175
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 338
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 338 are
12, 14, 16, . . . . 338
After observing the above list of the even numbers from 12 to 338 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 338 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 338
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 338
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 338
= 12 + 338/2
= 350/2 = 175
Thus, the average of the even numbers from 12 to 338 = 175 Answer
Method (2) to find the average of the even numbers from 12 to 338
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 338 are
12, 14, 16, . . . . 338
The even numbers from 12 to 338 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 338
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 338
338 = 12 + (n – 1) × 2
⇒ 338 = 12 + 2 n – 2
⇒ 338 = 12 – 2 + 2 n
⇒ 338 = 10 + 2 n
After transposing 10 to LHS
⇒ 338 – 10 = 2 n
⇒ 328 = 2 n
After rearranging the above expression
⇒ 2 n = 328
After transposing 2 to RHS
⇒ n = 328/2
⇒ n = 164
Thus, the number of terms of even numbers from 12 to 338 = 164
This means 338 is the 164th term.
Finding the sum of the given even numbers from 12 to 338
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 338
= 164/2 (12 + 338)
= 164/2 × 350
= 164 × 350/2
= 57400/2 = 28700
Thus, the sum of all terms of the given even numbers from 12 to 338 = 28700
And, the total number of terms = 164
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 338
= 28700/164 = 175
Thus, the average of the given even numbers from 12 to 338 = 175 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 1153
(2) What will be the average of the first 4339 odd numbers?
(3) Find the average of the first 2863 even numbers.
(4) Find the average of even numbers from 6 to 1496
(5) Find the average of odd numbers from 15 to 1281
(6) What will be the average of the first 4859 odd numbers?
(7) Find the average of the first 2585 even numbers.
(8) What is the average of the first 1482 even numbers?
(9) Find the average of the first 3611 odd numbers.
(10) Find the average of odd numbers from 13 to 1347