Question : Find the average of even numbers from 8 to 470
Correct Answer 239
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 470
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 470 are
8, 10, 12, . . . . 470
After observing the above list of the even numbers from 8 to 470 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 470 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 470
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 470
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 8 to 470
= 8 + 470/2
= 478/2 = 239
Thus, the average of the even numbers from 8 to 470 = 239 Answer
Method (2) to find the average of the even numbers from 8 to 470
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 470 are
8, 10, 12, . . . . 470
The even numbers from 8 to 470 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 470
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 8 to 470
470 = 8 + (n – 1) × 2
⇒ 470 = 8 + 2 n – 2
⇒ 470 = 8 – 2 + 2 n
⇒ 470 = 6 + 2 n
After transposing 6 to LHS
⇒ 470 – 6 = 2 n
⇒ 464 = 2 n
After rearranging the above expression
⇒ 2 n = 464
After transposing 2 to RHS
⇒ n = 464/2
⇒ n = 232
Thus, the number of terms of even numbers from 8 to 470 = 232
This means 470 is the 232th term.
Finding the sum of the given even numbers from 8 to 470
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 8 to 470
= 232/2 (8 + 470)
= 232/2 × 478
= 232 × 478/2
= 110896/2 = 55448
Thus, the sum of all terms of the given even numbers from 8 to 470 = 55448
And, the total number of terms = 232
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 8 to 470
= 55448/232 = 239
Thus, the average of the given even numbers from 8 to 470 = 239 Answer
Similar Questions
(1) Find the average of the first 2344 odd numbers.
(2) Find the average of the first 1100 odd numbers.
(3) What will be the average of the first 4893 odd numbers?
(4) Find the average of the first 2885 even numbers.
(5) Find the average of even numbers from 12 to 420
(6) What is the average of the first 508 even numbers?
(7) What will be the average of the first 4938 odd numbers?
(8) Find the average of even numbers from 10 to 924
(9) What is the average of the first 1946 even numbers?
(10) Find the average of odd numbers from 11 to 543