Question:
Find the average of even numbers from 10 to 478
Correct Answer
244
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 478
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 478 are
10, 12, 14, . . . . 478
After observing the above list of the even numbers from 10 to 478 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 478 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 478
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 478
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 478
= 10 + 478/2
= 488/2 = 244
Thus, the average of the even numbers from 10 to 478 = 244 Answer
Method (2) to find the average of the even numbers from 10 to 478
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 478 are
10, 12, 14, . . . . 478
The even numbers from 10 to 478 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 478
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 478
478 = 10 + (n – 1) × 2
⇒ 478 = 10 + 2 n – 2
⇒ 478 = 10 – 2 + 2 n
⇒ 478 = 8 + 2 n
After transposing 8 to LHS
⇒ 478 – 8 = 2 n
⇒ 470 = 2 n
After rearranging the above expression
⇒ 2 n = 470
After transposing 2 to RHS
⇒ n = 470/2
⇒ n = 235
Thus, the number of terms of even numbers from 10 to 478 = 235
This means 478 is the 235th term.
Finding the sum of the given even numbers from 10 to 478
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 478
= 235/2 (10 + 478)
= 235/2 × 488
= 235 × 488/2
= 114680/2 = 57340
Thus, the sum of all terms of the given even numbers from 10 to 478 = 57340
And, the total number of terms = 235
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 478
= 57340/235 = 244
Thus, the average of the given even numbers from 10 to 478 = 244 Answer
Similar Questions
(1) Find the average of the first 4089 even numbers.
(2) Find the average of the first 3426 even numbers.
(3) Find the average of even numbers from 10 to 406
(4) Find the average of odd numbers from 7 to 243
(5) Find the average of even numbers from 6 to 306
(6) Find the average of odd numbers from 15 to 565
(7) Find the average of even numbers from 10 to 474
(8) Find the average of the first 3051 even numbers.
(9) What is the average of the first 1742 even numbers?
(10) Find the average of odd numbers from 15 to 1627