Question:
Find the average of even numbers from 12 to 466
Correct Answer
239
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 466
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 466 are
12, 14, 16, . . . . 466
After observing the above list of the even numbers from 12 to 466 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 466 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 466
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 466
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 466
= 12 + 466/2
= 478/2 = 239
Thus, the average of the even numbers from 12 to 466 = 239 Answer
Method (2) to find the average of the even numbers from 12 to 466
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 466 are
12, 14, 16, . . . . 466
The even numbers from 12 to 466 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 466
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 466
466 = 12 + (n – 1) × 2
⇒ 466 = 12 + 2 n – 2
⇒ 466 = 12 – 2 + 2 n
⇒ 466 = 10 + 2 n
After transposing 10 to LHS
⇒ 466 – 10 = 2 n
⇒ 456 = 2 n
After rearranging the above expression
⇒ 2 n = 456
After transposing 2 to RHS
⇒ n = 456/2
⇒ n = 228
Thus, the number of terms of even numbers from 12 to 466 = 228
This means 466 is the 228th term.
Finding the sum of the given even numbers from 12 to 466
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 466
= 228/2 (12 + 466)
= 228/2 × 478
= 228 × 478/2
= 108984/2 = 54492
Thus, the sum of all terms of the given even numbers from 12 to 466 = 54492
And, the total number of terms = 228
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 466
= 54492/228 = 239
Thus, the average of the given even numbers from 12 to 466 = 239 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 1317
(2) Find the average of even numbers from 6 to 1028
(3) Find the average of the first 508 odd numbers.
(4) Find the average of odd numbers from 13 to 1215
(5) Find the average of the first 1990 odd numbers.
(6) Find the average of the first 339 odd numbers.
(7) Find the average of odd numbers from 9 to 1025
(8) Find the average of even numbers from 12 to 90
(9) Find the average of the first 2130 odd numbers.
(10) Find the average of the first 1344 odd numbers.