Question : Find the average of even numbers from 10 to 244
Correct Answer 127
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 244
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 244 are
10, 12, 14, . . . . 244
After observing the above list of the even numbers from 10 to 244 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 244 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 244
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 244
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 244
= 10 + 244/2
= 254/2 = 127
Thus, the average of the even numbers from 10 to 244 = 127 Answer
Method (2) to find the average of the even numbers from 10 to 244
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 244 are
10, 12, 14, . . . . 244
The even numbers from 10 to 244 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 244
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 244
244 = 10 + (n – 1) × 2
⇒ 244 = 10 + 2 n – 2
⇒ 244 = 10 – 2 + 2 n
⇒ 244 = 8 + 2 n
After transposing 8 to LHS
⇒ 244 – 8 = 2 n
⇒ 236 = 2 n
After rearranging the above expression
⇒ 2 n = 236
After transposing 2 to RHS
⇒ n = 236/2
⇒ n = 118
Thus, the number of terms of even numbers from 10 to 244 = 118
This means 244 is the 118th term.
Finding the sum of the given even numbers from 10 to 244
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 244
= 118/2 (10 + 244)
= 118/2 × 254
= 118 × 254/2
= 29972/2 = 14986
Thus, the sum of all terms of the given even numbers from 10 to 244 = 14986
And, the total number of terms = 118
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 244
= 14986/118 = 127
Thus, the average of the given even numbers from 10 to 244 = 127 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1978
(2) Find the average of even numbers from 6 to 176
(3) If the average of three consecutive even numbers is 14, then find the numbers.
(4) Find the average of odd numbers from 13 to 1229
(5) Find the average of even numbers from 4 to 1088
(6) What is the average of the first 1922 even numbers?
(7) What is the average of the first 824 even numbers?
(8) Find the average of even numbers from 10 to 814
(9) What is the average of the first 1535 even numbers?
(10) Find the average of the first 1159 odd numbers.