Question : Find the average of even numbers from 10 to 60
Correct Answer 35
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 60
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 60 are
10, 12, 14, . . . . 60
After observing the above list of the even numbers from 10 to 60 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 60 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 60
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 60
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 60
= 10 + 60/2
= 70/2 = 35
Thus, the average of the even numbers from 10 to 60 = 35 Answer
Method (2) to find the average of the even numbers from 10 to 60
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 60 are
10, 12, 14, . . . . 60
The even numbers from 10 to 60 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 60
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 60
60 = 10 + (n – 1) × 2
⇒ 60 = 10 + 2 n – 2
⇒ 60 = 10 – 2 + 2 n
⇒ 60 = 8 + 2 n
After transposing 8 to LHS
⇒ 60 – 8 = 2 n
⇒ 52 = 2 n
After rearranging the above expression
⇒ 2 n = 52
After transposing 2 to RHS
⇒ n = 52/2
⇒ n = 26
Thus, the number of terms of even numbers from 10 to 60 = 26
This means 60 is the 26th term.
Finding the sum of the given even numbers from 10 to 60
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 60
= 26/2 (10 + 60)
= 26/2 × 70
= 26 × 70/2
= 1820/2 = 910
Thus, the sum of all terms of the given even numbers from 10 to 60 = 910
And, the total number of terms = 26
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 60
= 910/26 = 35
Thus, the average of the given even numbers from 10 to 60 = 35 Answer
Similar Questions
(1) What is the average of the first 1716 even numbers?
(2) Find the average of the first 1899 odd numbers.
(3) What is the average of the first 210 even numbers?
(4) Find the average of odd numbers from 11 to 329
(5) Find the average of the first 3516 odd numbers.
(6) Find the average of odd numbers from 9 to 387
(7) Find the average of even numbers from 8 to 1154
(8) Find the average of odd numbers from 15 to 1691
(9) What is the average of the first 427 even numbers?
(10) Find the average of even numbers from 6 to 362