Question:
Find the average of even numbers from 10 to 110
Correct Answer
60
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 110
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 110 are
10, 12, 14, . . . . 110
After observing the above list of the even numbers from 10 to 110 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 110 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 110
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 110
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 110
= 10 + 110/2
= 120/2 = 60
Thus, the average of the even numbers from 10 to 110 = 60 Answer
Method (2) to find the average of the even numbers from 10 to 110
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 110 are
10, 12, 14, . . . . 110
The even numbers from 10 to 110 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 110
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 110
110 = 10 + (n – 1) × 2
⇒ 110 = 10 + 2 n – 2
⇒ 110 = 10 – 2 + 2 n
⇒ 110 = 8 + 2 n
After transposing 8 to LHS
⇒ 110 – 8 = 2 n
⇒ 102 = 2 n
After rearranging the above expression
⇒ 2 n = 102
After transposing 2 to RHS
⇒ n = 102/2
⇒ n = 51
Thus, the number of terms of even numbers from 10 to 110 = 51
This means 110 is the 51th term.
Finding the sum of the given even numbers from 10 to 110
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 110
= 51/2 (10 + 110)
= 51/2 × 120
= 51 × 120/2
= 6120/2 = 3060
Thus, the sum of all terms of the given even numbers from 10 to 110 = 3060
And, the total number of terms = 51
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 110
= 3060/51 = 60
Thus, the average of the given even numbers from 10 to 110 = 60 Answer
Similar Questions
(1) Find the average of the first 1035 odd numbers.
(2) Find the average of the first 4783 even numbers.
(3) Find the average of even numbers from 6 to 456
(4) Find the average of odd numbers from 13 to 601
(5) Find the average of the first 3926 even numbers.
(6) Find the average of odd numbers from 11 to 625
(7) Find the average of odd numbers from 3 to 525
(8) Find the average of odd numbers from 13 to 453
(9) Find the average of the first 2678 even numbers.
(10) What is the average of the first 415 even numbers?