Question:
Find the average of even numbers from 12 to 84
Correct Answer
48
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 84
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 84 are
12, 14, 16, . . . . 84
After observing the above list of the even numbers from 12 to 84 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 84 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 84
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 84
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 84
= 12 + 84/2
= 96/2 = 48
Thus, the average of the even numbers from 12 to 84 = 48 Answer
Method (2) to find the average of the even numbers from 12 to 84
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 84 are
12, 14, 16, . . . . 84
The even numbers from 12 to 84 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 84
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 84
84 = 12 + (n – 1) × 2
⇒ 84 = 12 + 2 n – 2
⇒ 84 = 12 – 2 + 2 n
⇒ 84 = 10 + 2 n
After transposing 10 to LHS
⇒ 84 – 10 = 2 n
⇒ 74 = 2 n
After rearranging the above expression
⇒ 2 n = 74
After transposing 2 to RHS
⇒ n = 74/2
⇒ n = 37
Thus, the number of terms of even numbers from 12 to 84 = 37
This means 84 is the 37th term.
Finding the sum of the given even numbers from 12 to 84
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 84
= 37/2 (12 + 84)
= 37/2 × 96
= 37 × 96/2
= 3552/2 = 1776
Thus, the sum of all terms of the given even numbers from 12 to 84 = 1776
And, the total number of terms = 37
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 84
= 1776/37 = 48
Thus, the average of the given even numbers from 12 to 84 = 48 Answer
Similar Questions
(1) Find the average of odd numbers from 3 to 243
(2) What will be the average of the first 4337 odd numbers?
(3) Find the average of the first 4061 even numbers.
(4) Find the average of odd numbers from 5 to 397
(5) Find the average of odd numbers from 7 to 1455
(6) What is the average of the first 1521 even numbers?
(7) Find the average of the first 2818 even numbers.
(8) What will be the average of the first 4685 odd numbers?
(9) Find the average of odd numbers from 11 to 975
(10) Find the average of odd numbers from 9 to 995