Question : Find the average of even numbers from 4 to 220
Correct Answer 112
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 220
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 220 are
4, 6, 8, . . . . 220
After observing the above list of the even numbers from 4 to 220 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 220 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 220
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 220
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 4 to 220
= 4 + 220/2
= 224/2 = 112
Thus, the average of the even numbers from 4 to 220 = 112 Answer
Method (2) to find the average of the even numbers from 4 to 220
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 220 are
4, 6, 8, . . . . 220
The even numbers from 4 to 220 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 220
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 4 to 220
220 = 4 + (n – 1) × 2
⇒ 220 = 4 + 2 n – 2
⇒ 220 = 4 – 2 + 2 n
⇒ 220 = 2 + 2 n
After transposing 2 to LHS
⇒ 220 – 2 = 2 n
⇒ 218 = 2 n
After rearranging the above expression
⇒ 2 n = 218
After transposing 2 to RHS
⇒ n = 218/2
⇒ n = 109
Thus, the number of terms of even numbers from 4 to 220 = 109
This means 220 is the 109th term.
Finding the sum of the given even numbers from 4 to 220
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 4 to 220
= 109/2 (4 + 220)
= 109/2 × 224
= 109 × 224/2
= 24416/2 = 12208
Thus, the sum of all terms of the given even numbers from 4 to 220 = 12208
And, the total number of terms = 109
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 4 to 220
= 12208/109 = 112
Thus, the average of the given even numbers from 4 to 220 = 112 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 150
(2) What is the average of the first 58 even numbers?
(3) Find the average of even numbers from 10 to 424
(4) Find the average of even numbers from 4 to 292
(5) Find the average of even numbers from 4 to 88
(6) Find the average of odd numbers from 9 to 517
(7) Find the average of the first 3068 even numbers.
(8) Find the average of even numbers from 4 to 1858
(9) Find the average of the first 3913 odd numbers.
(10) Find the average of the first 1073 odd numbers.