Question : Find the average of even numbers from 10 to 364
Correct Answer 187
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 364
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 364 are
10, 12, 14, . . . . 364
After observing the above list of the even numbers from 10 to 364 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 364 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 364
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 364
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 10 to 364
= 10 + 364/2
= 374/2 = 187
Thus, the average of the even numbers from 10 to 364 = 187 Answer
Method (2) to find the average of the even numbers from 10 to 364
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 364 are
10, 12, 14, . . . . 364
The even numbers from 10 to 364 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 364
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 10 to 364
364 = 10 + (n – 1) × 2
⇒ 364 = 10 + 2 n – 2
⇒ 364 = 10 – 2 + 2 n
⇒ 364 = 8 + 2 n
After transposing 8 to LHS
⇒ 364 – 8 = 2 n
⇒ 356 = 2 n
After rearranging the above expression
⇒ 2 n = 356
After transposing 2 to RHS
⇒ n = 356/2
⇒ n = 178
Thus, the number of terms of even numbers from 10 to 364 = 178
This means 364 is the 178th term.
Finding the sum of the given even numbers from 10 to 364
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 10 to 364
= 178/2 (10 + 364)
= 178/2 × 374
= 178 × 374/2
= 66572/2 = 33286
Thus, the sum of all terms of the given even numbers from 10 to 364 = 33286
And, the total number of terms = 178
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 10 to 364
= 33286/178 = 187
Thus, the average of the given even numbers from 10 to 364 = 187 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 973
(2) Find the average of odd numbers from 9 to 59
(3) Find the average of odd numbers from 3 to 681
(4) Find the average of odd numbers from 13 to 839
(5) What will be the average of the first 4406 odd numbers?
(6) What is the average of the first 338 even numbers?
(7) Find the average of odd numbers from 11 to 227
(8) Find the average of the first 416 odd numbers.
(9) Find the average of odd numbers from 15 to 663
(10) Find the average of even numbers from 6 to 1784