Question : Find the average of even numbers from 12 to 118
Correct Answer 65
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 118
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 118 are
12, 14, 16, . . . . 118
After observing the above list of the even numbers from 12 to 118 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 118 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 118
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 118
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 118
= 12 + 118/2
= 130/2 = 65
Thus, the average of the even numbers from 12 to 118 = 65 Answer
Method (2) to find the average of the even numbers from 12 to 118
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 118 are
12, 14, 16, . . . . 118
The even numbers from 12 to 118 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 118
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 118
118 = 12 + (n – 1) × 2
⇒ 118 = 12 + 2 n – 2
⇒ 118 = 12 – 2 + 2 n
⇒ 118 = 10 + 2 n
After transposing 10 to LHS
⇒ 118 – 10 = 2 n
⇒ 108 = 2 n
After rearranging the above expression
⇒ 2 n = 108
After transposing 2 to RHS
⇒ n = 108/2
⇒ n = 54
Thus, the number of terms of even numbers from 12 to 118 = 54
This means 118 is the 54th term.
Finding the sum of the given even numbers from 12 to 118
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 118
= 54/2 (12 + 118)
= 54/2 × 130
= 54 × 130/2
= 7020/2 = 3510
Thus, the sum of all terms of the given even numbers from 12 to 118 = 3510
And, the total number of terms = 54
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 118
= 3510/54 = 65
Thus, the average of the given even numbers from 12 to 118 = 65 Answer
Similar Questions
(1) Find the average of the first 768 odd numbers.
(2) Find the average of the first 3148 even numbers.
(3) Find the average of the first 3320 even numbers.
(4) What is the average of the first 1721 even numbers?
(5) Find the average of even numbers from 12 to 1206
(6) Find the average of the first 4682 even numbers.
(7) Find the average of odd numbers from 11 to 277
(8) Find the average of the first 1892 odd numbers.
(9) Find the average of odd numbers from 5 to 81
(10) What is the average of the first 1645 even numbers?