Question:
Find the average of even numbers from 12 to 208
Correct Answer
110
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 208
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 208 are
12, 14, 16, . . . . 208
After observing the above list of the even numbers from 12 to 208 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 208 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 208
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 208
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 208
= 12 + 208/2
= 220/2 = 110
Thus, the average of the even numbers from 12 to 208 = 110 Answer
Method (2) to find the average of the even numbers from 12 to 208
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 208 are
12, 14, 16, . . . . 208
The even numbers from 12 to 208 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 208
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 208
208 = 12 + (n – 1) × 2
⇒ 208 = 12 + 2 n – 2
⇒ 208 = 12 – 2 + 2 n
⇒ 208 = 10 + 2 n
After transposing 10 to LHS
⇒ 208 – 10 = 2 n
⇒ 198 = 2 n
After rearranging the above expression
⇒ 2 n = 198
After transposing 2 to RHS
⇒ n = 198/2
⇒ n = 99
Thus, the number of terms of even numbers from 12 to 208 = 99
This means 208 is the 99th term.
Finding the sum of the given even numbers from 12 to 208
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 208
= 99/2 (12 + 208)
= 99/2 × 220
= 99 × 220/2
= 21780/2 = 10890
Thus, the sum of all terms of the given even numbers from 12 to 208 = 10890
And, the total number of terms = 99
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 208
= 10890/99 = 110
Thus, the average of the given even numbers from 12 to 208 = 110 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1470
(2) Find the average of the first 609 odd numbers.
(3) What is the average of the first 802 even numbers?
(4) Find the average of the first 4512 even numbers.
(5) Find the average of even numbers from 10 to 262
(6) What is the average of the first 604 even numbers?
(7) Find the average of odd numbers from 3 to 969
(8) Find the average of the first 3489 odd numbers.
(9) Find the average of the first 3269 even numbers.
(10) Find the average of odd numbers from 7 to 217